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i 



WINSLO WS 



COliPEEHENSIYE MATHEMATICS: 



BEING AN EXTENSIVE CABINET OF 



KUMERICAL, ARITHMETICAL, AND :MATHEMATICAL FACTS, 
TABLES, DATA, FORMULAS, AND PRACTICAL RULES FOR 
THE GENERAL BUSINESS-MAN, MERCHANT, MECHANIC, 
ACCOUNTANT, TEACHERS OF SCHOOLS, GEOME- 
TRICIANS, AND SCIENTISTS; APPROPRIATELY 
ARRANGED AND APPLIED. 



BY 

E. S. WIN SL O W. 



Sixth Edition, Enlarged and Improved, 



BOSTON: 

PUBLISHED BY THE AUTHOR. 

186 7. 

/ 



G1S,0, W. LTNDSEY, Local Agent for the sale of Winslow's Mathematical 
Works, No. 897 Washington Street, Boston. 



Entered, according to Act of Congress, in the year 1867, by 

E. S. WIN SLOW, 

In the Clerk's Office of the District Court of the District of Massachusetts. 



Stkbeotyped by C. J. Petebs & Soir. 



Frinted by Wm. A. HaU, No. 46, Congress Street, Boston. 



I 



PREEACE 
TO THE COMPEEHENSIVE MATHEMATICS. 



On presenting this work to the public, it may be proper to state 
that it has been designed and written mainly for the practical 
man. It contains a vast array of Numerical, Arithmetical, and 
Mathematical facts, tables, datai, formulas, and rules, pertaining 
to a great variety of subjects, and applicable to a diversity of ends, 
as well as much information of a more general nature, valuable to 
the artisan, and commercial classes ; thus meeting the wants, in an 
eminent degree, of the lovers of the exact sciences, and the prac- 
tical wants of students in the mathematics. 

The facts and data alluded to have been gathered, with much 
care and patience, from a great variety of sources, or derived, often 
by toilsome investigations, from known and accredited truths. 
The care that has been taken in respect to these, it is thought, 
should secure for this particular department reliance and trust. 

The tables, which are numerous, have, with few exceptions, been 
composed and arranged expressly for the work, and a confidence 
is felt that they may be relied on for accuracy. 

From the valuable works of Dr. Ure, Adcock, Gregory, Grier, 
Brunton ; from the publications of the transactions of London, 
Edinburgh, and Dublin Philosophical Societies ; and from the pub- 
lications by the Smithsonian Institute, much valuable information 
has been gained, relating mainly to machinery and the arts ; and 
to these sources the author feels indebted. 

The conciseness with which the work has been generally written 
would, perhaps, be found an objection, were it not that all the pro- 
positions and problems of intricacy are accompanied with exam- 
ples and illustrations, and, in the matters of Geometry, additionally 
accompanied with diagrams. The whole, it is thought, will appear 
clear to him who consults it. A prominent feature in the design 
has been to produce a useful work, and one which in the way of 
price shall be readily accessible to all. 

iii 



iv ~ PREFACE. 

PRETA E 

TO THE FOREIGN AND DOMESTIC COMMERCIAL 
CALCULATOR. 



This work is composed of the first four sections of the author's 
" Comprehensive Mathematics." It was thought advisable 
to publish this portion of that work in a separate form on account 
of price ; more especially as it contains all of a commercial nature 
treated of in that work. Indeed, the contents of that work were 
arranged expressly to this end. The Table of Contents in both 
works is the same. The work being stereotyped, this could not well 
be avoided. The Table of Contents, therefore, in either work, is 
that of the " Comprehensive Mathematics," and the first 
four, sections thereof; that is, Section I., Section II., Section III., 
and Section A., is that of the " Foreign and Domestic Commer- 
cial Calculator." 



P K E F A C E 

TO THE TIN-PLATE AND SHEET-IRON WORKERS' MONITOR. 



This work is composed of Section VI. of the author's " Com- 
prehensive Mathematics," with portions of other sections of 
that work. It embraces all that is contained in the last-mentioned 
•work of special interest to the Tinsmith, as such. It may be re- 
lied on for accuracy in all particulars, and is believed to be the 
first and only reliable work of the kind ever published. It is pub- 
lished in separate form on account of price, and with the view of 
affording apprentices and students every possible facility of obtain- 
ing it. It contains over 100 pages, nearly 50 diagrams, and step- 
by-step directions for constructing, mecJianicallij, not less than 30 
unhke and different patterns, embracing all of the more difficult 
and complicated in use, and s.everal of new and beautiful designs. 



CONTENTS. 



SECTION A. 

PAGE 

Foreign Moneys of Account ... a 1 
Foreign Linear and Surface Meas- 
ures a 14 

Foreign Weights a 2'.i 

Foreign Lifxuid Measures . . . . a 35 

Foreign Dry Measures « 44 

Custom House Allowances on Du- 
tiable Goods, &c « 51 

Table of Established Tares . . . a 52 



SECTION I. 



moneys of account, coins, 
weights, and measures of 
the united states; foreign 
gold coins, &c. 

Explanations of Signs .... 12 

Moneys of Account of the United 

States 13 

Comparative Value of Gold and 

Silver 13 

Gold, pure ; value of, by weight . . 15 
Mint Gold, Standard of, &c. . . . 15 
Gold Coins, their weights and val- 
ues 15 

Silver, pure ; value of, by weight . IG 
Mint Silver, Standard ot, &c. ... 16 
Silver Coins, their weights and 

values 16 

Copper Coins, &c 16 

Tijesent Par Value of Silver Coins 

issued prior to June, 1853 .... 17 
Currencies of the ditferent States 

of the Union 17 

The Metrical System of Weights 

and Measures 18 

Foreign Gold Coins, Tables of, &c. 19 
Foreign Silver Coins, Values of, 25 

1* 



WEIGHTS AND MEASURES. 

PAGE 

Long or Linear Measure ... 25 

Cloth Measure 25 

Land Measure 25 

Engnieer's Chain 25 

Shoemaker's Measure 26 

Miscellaneous Measures 26 

Square or Superficial Meas- 
ure 26 

Measure for Land 26 

Circular Measure 27 

Cubic or Solid Measure ... 27 
General Measure of Weight, 28 

Gross Weight 28 

Troy Weight 28 

Apothecaries' Weight, 28 

Diamonds, Measure of Value, &c., 28 

Liquid Measure 28 

Imperial Liquid Measure 29 

Ale Measure 29 

Dry Measure 29 

Imjjerial Dry Measure 30 



SECTION IL 



ailSCELLANEOUS FACTS, CALCU- 
LATIONS, AND MATHEMATICAL 
DATA. 

Specific Gravities, Tables of, 31 

Weight per Bushel of Articles . . 35 
Weightper Barrel of Articles . . . 35 
Weights of different Measures of 

various Articles 35 

Weight of Coals, &c., Tables . 35, 55 
Practical Approximate Weight in 

Pounds of Various Articles ... 36 



Ropes and Cables . 

V 



36 



VI 



CONTENTS. 



PAOE 

Weight and Strength of Iron 
Chains 37 

Comparative Weight of Metals, 
Table 38 

Weight of Rolled Iron, Square Bar, 
Tables 38 

Weight of Various Metals, differ- 
ent Forms of Bar 39 

Weight of Round-rolled Iron, Ta- 
ble 40 

Weight of Cast-iron Prisms of dif- 
ferent forms, &c 40 

Weight of Flat-rolled Iron, Table, 4'i 

Weight of Different Metals,in Plate, 44 

The American Wire Gauge . 45 

The Values of the Nos. American 
Wire Gauge and iiirmingham 
Wire Gauge, in the United .States, 

inch, Tables of 45 

The Numl)er of Linear Feet in a 
l*ound of different kinds of Wire 
of dilfcrent Sizes, Table of, &c.. 40 
Characteristics, &c., of Alloys of 

Copper and Zinc,— Brass . ... -47 
The Weight per.s<iuaie Foot of dif- 
ferent Rolled Metals of different 
tliicknesses by the Wire Gauge, 

Table 4S 

Tin Plates, Sizes, &c.. Table. 4'J 
Sheet Iron, Sheet Zinc, Copper 
Sheathing, Yellow jNletal, Weight 

of, &c 4'J 

Capacity in Gallons of Cylindrical 

cans, &c., Table 50 

Weight of IMpes 5.i 

Weight of Pipes, Table 53 

Weight of Cast-iron and Lead 

Balls 54 

Weight of Hollow Balls or Shells, 54 

Analysis of Coals 55 

Weight, Heating Power, »^c., of 
Coals and other kinds of Fuel, 
Table 55 

Mensuration of Lumber . . . 5G 

BoArd Measure 50 

To 3Ieasure Square Timber .... 50 

To Measure Round Timber .... 50 

Table relative to the 3Ieasuremeut 
of Round Timber 57 

TO find the Solidity of the greatest 
Rectangular Stick that can be cut 
from a Lo<^ of Given Dimensions, 58 

To find the Solidity of the greatest 
Square Stick that can be cut from 
a Round Stick of Given Dimen- 
sions 69 

To find the Contents of a Log in 
Board Measure 59 

Gauging 60 



To find the Dimension'? of Vessels 
of different Forms, for holding 
Given Quantities 62 

Cask Gauging, all Forms of 
Casks . . . 63 

To find the Contents of a Cask, the 
same as would be given by the 
Gau/^ing Rod 66 

To find the Diagonal and Length 
of a Cask 66 

Ullage 67 

To find the Ullage of a Standing 
Cask 67 

To find the Ullage when the Cask is 
upon its Bilge 67 

To find the.l^iiantitv of Liquor in a 
Ca><k by its Weight 68 

Customary Rule by Freighting Mer- 
chants for finding the Cubic 
Measurement of Casks 68 

Tonnage of Vessels, to Calcu- 
late 69 

Ok Conduits, or Pipes 70 

To find the re<pusite thickness of a 
Pipe to support a Given Head of 
Water 70 

To fin<l the ^'elo^ity of Water pass- 
ing tlirough a Pipe 71 

To find the Hea<l of Water requi- 
site to a Ue(piired Velocity 
through a Pipe 71 

To find the (Quantity of Water Dis- 
charged by a Pipe in a Given 
Time 71 

To find the Specific Gravity of a 
Body lu'avier than Water .... 72 

To find the Si)ecific (Gravity of a 
Body lighter than Water .... 72 

To find tlie Specific Gravity of a 
Fluid 72 

To find the (Quantity of each of the 
several Metals composing an Al- 
loy 72 

To find the Lifling-power of a Bal- 
loon 73 

To find the Diameter of a Balloon 
e(|ual to the Raising of a Given 
Weight 73 

To find the Thickness of a Hollow 
Metallic Globe that shall have a 
Given Buoyancy in a Given 
Liquid 73 

To Cut a Square Sheet of Metal so 
as to form a Vessel of the Great-, 
est Capacity the Sheet admits of . 73 

Comparative Cohesive Forces of 
Substances, Table 7-1 

Alloys having a Tenacity greater 
than the Sum of their Con- 
stituents 74 



CONTENTS. 



Vll 



PAGE 

Alloys having a Density greater 
tlian the Mean of their Con- 
stituents 75 

Alloys liaving a Density less than 

the Mean of their Constituents . 75 
Relative Powers of different Metals 

to Conduct lilectricity 75 

Dilations of Solids bv Heat, Table 75 
Melting Points of Metals and other 

Substances, Tablk 76 

llelative Powers of Substances to 

Radiate Heat, Table 76 

Boiling Points of Fluids 76 

Freezing Points of Fluids .... 77 
iCxpansion of Fluids by Heat . . . 77 
Relative Powers of Substances to 

Conduct Heat 77 

Ductility and Maleability of Metals, 77 
Quantity per cent, of Nutritious 
Matter contained in different Ar- 
ticles of Food 78 

Standard, &c., of Alcohol 78 

Quantity per cent, of Absolute Al- 
coliol contained in different Pure 
Liquors, Wines, &c., Tajjle . . 7S 
Proof of Spirituous Liquors ... 78 
Comparative Weight of Timber in 
a Green and Seasoned State, Ta- 
ble, &c 79 

Relative Power of different kinds of 

Fuel to Produce Heat, Table, . 79 
Relative Illuminating Power of dif- 
ferent Materials, Table and Re- 
marks, ...... 80 

Thermometers, different kinds, 
to Reduce one to another, &c., . 82 

Horse-Power 83 

Animal Power 83 

STEA^r, Tables in relation to, 

&c., S3, 308 

Velocity and Force of AYind, Ta- 
ble 84 

Curvature of the Earth . . . .84,213 
Degrees of Longitude, Lengths of, 

&c., 84 

Time, with respect to Longitude, 84 

Velocity of Sound 84 

Velocity of Light 85 

Gravitation 85, 302 

Area of the Earth, its Density, &c., 85 

Chemical Elements 86 

Elementary Constituents of Bodies, 

Table 87 

Combinations by Weight of the 
Gases in forming Compounds, 

Table 87 

Combinations by Volume of the 
Gases, their Condensation, &c., 

in forming Compounds 89 

Atomic Weight 89 



PAOK 

Chemical and other Properties of 
Various Substances 90 



SECTION III. 



practical arithmetic. 

Vulgar Fractions 95 

Reduction of Vulgar Fractions . . 95 
Addition of Vulgar Fractions . . . 99 
Subtraction of Vulgar Fractions . 99 
Division of Vulgar Fractions . . . 100 
Multiplication of Vulgar Fractions 100 
3Iultiplication and Division of 
Fractions Combined 101 

Cancellation 96, 97, 102 

To Reduce a Fraction in a higher, 
to an equivalent in a given Tow- 
er denomination 102 

To Reduce a Fraction in a lower, 
to an equivalent in a given high- 
er denomination 102 

To Pediice a Fraction to Whole 
iS^'umbers in lower given denom- 
inations 103 

To Reduce Fractions in lower de- 
nominations to given liigher de- 
nominations 103 

To work Vulgar Fractions by the 
Rule of Three, or Proportion . . 104 

Decimal Fractions 104 

Addition of Decimals 105 

Subtraction of Decimals 105 

^lultiplication of Decimals .... 106 

Division of Decimals 106 

Reduction of Decimals 107 

To work Decimals by the Rule of 

Three 108 

Proportion, or Rule of Three ... 109 

Compound Proportion 110 

Conjoined Proportion, or Chain 
Rule 112 

Percentage 114 

Interest 120 

Compound Interest 122 

Bank Interest, or Bank Discount . 127 

Discount 129 

Compound Discount 129 

Profit and Loss 130 

Equation of Payments 132 

General Average 134 

Assessment of Taxes 136 

Insurance 1.36 

Life Insurance • 136 

Fellowship 138 



vni 



CONTENTS. 



PAOS 

Alligation 139 

Involution 141 

Evolution 141 

To Extract the Square Root ... 142 
To Extract the Cube Root .... 143 

To P^xtract any Root 145 

Arithmetical Progression 140 

Geometrical Progression 150 

Annuities 154 

Of Installments generally . . . . 1C4 

Permutation 100 

Combination 107 

Problems 109 



SECTION IV. 
geometry. 
Definitions, Construction of 

EKiURES, &C 172 

To Bisect a Line 170 

To Erect a Perpendicular 176 

To Let Eall a Perjx'iulicular . . . 170 
To Erect a Perpendicular on the 

end of a Line 177 

To draw a Circle through any three 
points not in a straight line, and 
to tind the Centre of a Circle, or 

Arc 177 

To tind the Length of an Arc of a 
a Circle approximately by Me- 
chanics 177 

From a given l*oint to draw a 

Tangent to a Circle 177 

To draw from or to the Circumfer- 
ence of a Circle, lines tending 
to the Centre, when the latter is 

inacccHsible 177 

To describe an Oval Arch on a 

given Conjugate Diameter ... 178 
To describe an Oval of a given 

Length and Breadth 178 

To describe an Arc or Segment of 

a Circle of Large Radius .... 179 
To describe an Oval Arch, the 
Span and Rise being given . . . 179 

Gothic Arches, to draw 180 

Polygons, to construct 181 

Polygons, to inscribe in a given 

Circle 181 

Polygons, to circumscribe about a 

given Circle 181 

To produce a Square of the same 

Area as a given Triangle .... 181 
To construct a Parabola . . . 182, 355 
To Construct a Hyperbola . . 182, 319 
To bisect any given Triangle ... 182 



PIOS 

To draw a Triangle equal in Area 
to two given Triangles 183 

To describe a Circle equal in Area 
to two given Circles 183 

To construct a Tothed, or Cog- 
wheel 183 

Of the Conic Sections .... 184 

Mensuration of Lines and Super- 
ficies. 

Triangles 185 

Of Right-Angled Triangles ... 186 
Of Oblique-Angled Triangles . . 187 
To find the Area of a Triangle . 188 
To tind the Hypotenuse of a Tri- 
angle 189 

To tind the Base, or Perpendicu- 
lar, of a Triangle 188, 189 

To tind the Height of an inacces- 
sible Object 189 

To lind the Distance of an inac- 
cessible Object 190 

To find the Area of a Scjuare, 
Rectangle, Rhombus, or Rhom- 
boid 190 

To find the Area of a Trapezoid . 191 
To find the Area of a Trapezium . 191 
Of Polygons, Table, &c. ... Ii4 
To find the l*erpendicular of a 
Rhombus, Rhomboid, or Trape- 
zoid 192 

To tind the Diagonal of a Rhom- 
bus, Rhomb()i<l, or Trapezoid . 192 
To find the Area of a regular or 

irregular Polygon 195 

Circle 196 

Tiie Circle and its Sections .... 197 

To llnd the Diameter, Circumfer- 
ence, and Area of a Circle . . . 198 

To lincl the Length of an Arc of a 
Circle 199 

To tind the Area of a Sector of a 
Circle 201 

To find the Area of a Segment of 
a Circle 201 

To find the Area of a Zone . . . 202 

To find the Diameter of a Circle of 
which a given Zone is a part . . 202 

To find the Area of a Crescent . . 202 

To find the Side of a Square that . 
shall contain an Area ecjual to 
that of a given Circle 202 

To find the Diameter of a Circle 
that shall have an Area equal to 
that of a given Square ..... 20'3 

To find the Diameters of three 
equal circles the greatest that 
can be inscribed in a given Cir- 
cle 202 



CONTENTS. 



IX 



PAGE 

To find the Diameters of four equal 
circles the greatest that can be 
inscribed in a given Circle . . . 202 
To find the Side of a Square in- 
scribed in a given Circle .... 203 
To find the Diameter of a Circle 
that will circumscribe a given 

Triangle 203 

To find the Diameter of the great- 
est Circle tliat can be inscribed 

in a given Triangle 203 

To divide a Circle into any num- 
ber of Concentric Circles of 

equal Areas 204 

To find the Area of the space con- 
tained between two Concentric 

Circles 205 

Ellipse 205 

To find the Area of an Ellipse . . 207 
To find the Length of the Circum- 
ference of an Ellipse ..... 207 
To find the Area of an Elliptic Seg- 
ment 207 

Parabola 209 

To find the Area of a Parabola. . 210 
To find the Area of a Zone of a 

Parabola 210 

To find the Altitude of a Parabola, 210 
To find the Length of a Semi-para- 
bola 210 

Hyperbola 211 

To find the Length of a Semi- 
hyperbola 212 

To find the Area of a Hyperbola . 212 

Cycloid, and Epicycloid ... 212 

To find the Length of the Curve of 

a Cycloid 213 

To find the Area of a Cycloid. . . 213 
To find the Distance of Objects at 

Sea, &c 213 

Stereometry, or Mensuration 
of Solids. 

Of Prisms 214 

Of Right Prisms or Cubes .... 215 
Of Parallelopipedons .... 215 

Of Pyramids 215 

Of Frustums of Pyramids . . 216 

Of Prismoids 210 

Of the Wedge 217 

Of Cylinders 217 

To find the Length of a Helix . . 217 

Of Cones 218 

Of Frustums of Cones. ... 65, 218 
Of Spheres or Globes .... 219 



PAGE 

Of Spherical Segments 219 

Of Spherical Zones 220 

To find the greatest Cube that can 

be cut from a given Sphere . . . 220 

Of Spheroids 221 

Of Segments of Spheroids .... 221 
Of the Middle Frustum of a Spher- 
oid 65,221 

Of Spindles 222 

Of the Middle Frustum of a Para- 
bolic Spindle 65, 222 

Of Parabolic Conoids . . . 65, 223 

Of Hyperboloids 223 

To find the Surface of a Cylindri- 
cal Ring 224 

To find the Solidity of a Cylindri- 
cal Ring 224 

Of the regular bodies .... 225 
Promiscuous Examples in 

Geometry 226 

Trigonometry 231 

Tables of Sines, Cosines, Tan- 
gents, &c 241 

Tables of Squares, Cubes, 

Square and Cube Roots, &c. 245 



SECTION V. 

mechanical powers, mechani- 
cal centres, circular mo- 
tion, strength of materi- 
als; STEAM, the steam EN- 
GINE, etc. 

The Lever 271 

The Wheel and Axle .... 272 

The Pulley 273 

The Inclined Plane 274 

The Wedge . 275 

The Screw 275 

Transverse Strength of Bodies . 279 

Deflections of Shafts, &c 286 

Resistance of Bodies to Tortion . 287 
Resistance of Bodies to Compres- 
sion 289 

Centres of Surfaces 291 

Centres of Solids 293 

Centres of Oscillation and 

Percussion 294 

Centre of Gyration 298 

Central Forces 300 



CONTENTS. 



PAOK 

Flywheels 301 

The Governor 301 

Force of Gra^^ty 302 

To find the Height of a Stream 

projected vertically from a Pipe, 303 
To find the Power requisite to 
project a Stream to any given 

Height 303 

Of Pendulums 301 

Screw-Cutting in a Lathe . . 305 
Table of Change Wheels for 

Screw-Cutting in a Lathe . . . 308 
Of Steam and the Steam En- 
gine 308 

Velocity of l*rojectile«, &c . . . . 313 
Steam, acting expansively . . . . 31.*i 
Of the Eccentric in a Steam En- 
gine 314 

Of Continuous Circular 3Io- 

tion 314 

To find the number of Revolu- 
tions made by the last, to one 
revolution of the tirst, in a train 
of Wheels and Pinions .... 315 
The distance from Centre to Cen- 
tre of two Wheels to work in 
contact given, and tlie ratio of 
Velocity between them, to find 
their Requisite Diumrtirs . . . 317 
To find the Velocity of a licit . . 317 
To find the Drall on a ^lachine . 317 
To find the Revolutions of the 

Throstle Spindle 318 

To find the Twist given to the 

Yarn by the Throstle 318 

Tkkth OF Wheels, &c 318 

To construct a Tooth, &c 319 

To find the Horse-Power of a 

Tooth 310 

Journals of Shafts 3'JO 

Hydrostatics 3J0 

Hydraulics 322 

Water- Wheels 323 

To find the Power of a Stream . . 324 
To construct a Water- Wheel to a 

Given Power and Fall 325 

Dynamics 32G 

Hydrostatic Press 326 



SECTION VL 

coverings of solids, or prob- 
lems in pattern cutting. 



Rk»larksand Definitions . 



.32 



PAOf 

To construct a Pattern for the* 
Lateral Portion of a vessel in the 
form of a Frustum of a Cone of 
given diameters and depth . . . 329 

To construct a Pattern for the 
Body of a vessel in the form of a 
Frustum of a Cone of given di- 
mensions, without plotting the 
dimensions 332 

To construct a I*attern for the 
Lateral Portion of a Flaring Ves- 
sel of given symmetry of outline 
and given capacity 333 

Table of IfELATivb: Propor- 
tions, Chords, &c 333 

The special tabular figure, the di- 
ameter of one end, and the Tubic 
Capacity of the vessel l>eing 
given, to find the diameter of 
th»' other end 3.30 

To construct a Pattern for tlie body 
of a Flaring Vessel of given 
tubular outline, and given dimen- 
hions, without plotting the di- 
mensions 338 

The Capacity in gallons of a ressel 
in the form of a Frustum of a 
Cone being given, and any two 
of its dimensions, to find the 
other <linunsion 310 

To construct I'atterns for flaring 
oval vessels of diirerent eccentri- 
cities and given dimensions, Nos. 
1,2,3 342 

To describe the bases for Nos.1,2,3,343 

Of Cylindrical Elbows. . . . 348 

To construct a Pattern for a Right- 
angled Cylindrical Elbow . . . .340 

To construct Oblique-angled El- 
bows 352 

To constnict Right-angled Elliptic 
Elbows 353 

To construct Oblicpie-angled Ellip- 
tic Elbows 353 

To construct Right Semi-hyperbo- 
las by intersecting lines . . 349, 353 

To construct the (Quadrant of a Cir- 
cle by intersecting lines .... 354 

To construct the (Quadrant of a 
given Ellipse by intersecting 
lines 354 

To construct the Quadrant of a Cy- 
cloidal Ellipse by intersecting 
lines 354 

To describe an Ellipse of given di- 
mensions by means of two Posts, 
a Pencil, and a String 354 

To find the length of the circum- 
ference of a given Ellipse . . 207, :i5i> 

To construct a Semi-parabola by 
intersecting lines 355 



CONTENTS. 



PAGE 

Ovals, to describe . 178, 343, 346, 347 

Of Circulak Elbows 355 

Table applicable to Circular El- 
bows 356 

To construct a Right-angled Circu- 
lar Elbow of 3, 4, 5, 6, 7, or 8 

pieces, &c .355 

To construct a Collar for a Cylin- 
drical Pipe of the same dianaeter 

as the receiving pipe 359 

To construct a Cylindrical Collar 
of a given Diameter to fit a Ke- 
ceiving-pipe of a greater given 

Diameter 360 

To construct a Cylindrical Collar 
to tit an Elliptic-cylinder at ei- 
ther right section of the El- 
lipse 361 

To construct a Cylindrical Collar 
of a given Diameter, to fit a Cyl- 
inder of the same Diameter, at 
any given Angle to the side of 

the Cylinder 361 

To construct a Cylindrical Collar, 
or Spout, of a given Diameter, 
to fit a Cylinder of a greater giv- 
en Diameter, at a given Angle 
to the side of the Cylinder . . . 362 
Of Spouts for Vessels .... 363 
Of Pitched or Bevelled Covers . . 364 
To construct a Bevelled Circular 
Cover of a given Rise and giv- 
en Diameter 364 



PAOE 

To construct a Pattern for a Bev- 
elled Elliptical Cover of a given 
Rise to fit an Elliptic Boiler of 
given Diameters 365 

To construct a Bevelled Cover of a 
given Rise, to fit a False-Oval 
Boiler of given length and width 365 

Of Cax-tops 366 

To construct a Can-top of a given 
Depth and given Diameters . . 366 

To construct a Can-top of a given 
Pitch, and given Diameters . . 367 

Of Lips for Measures .... 368 

To construct a Lip for a Measure, 
the Diameter of the Top of the 
Measure being given 369 

Of Sheet Pans 369 

To cut the Corners for a Perpen- 
dicular-sided Sheet Pan .... 370 

To cut the Corners for an Oblique- 
sided Sheet Pan 370 

To construct a Heart, or Heart- 
shaped Cake-Cutter 370 

To construct a Mouth-piece for a 
Speaking-Tube 370 

To construct a Pattern for the 
Body of a Circular - bottomed 
Flaring Coal-Hod, all the curv es 
to be arcs of circles 371 

Solders, Alloys, and Compo- 
sitions 373 



DEFINITIONS 

OF THE SIGNS USED IN THE FOLLOWING WORK. 



= Equal to. The si^n of equality ; as 16 oz. ^ 1 lb. 

-{- Plus, or More, The sign of addition ; as 8 + 12 = 20. 

— Minus, or Less, The sign of subtraction ; as 12 — 8 = 4. 

X Multiplied by. The sign of multiplication ; as 12 X 8 = 96. 

-^ Divided by. The sign of division ; as 12 -^ 4 = 3. 

>r Difference between the given numbers or quantities ; thus, 12 ^ 8, or 
8 <y^ 12, shows that the less number is to be subtracted from the 
greater, and the difference, or remainder, only, is to be used ; so, • 
too, height j^ breadth, shows that the difference between the height 
and breadth is to be taken. 

: :: : Proportion ; as 2 : 4 :i 3 I 6 ; that is, as 2 is to 4, so is 3 to 6. 

>v/ Sign of the square root; prefixed to any number indicates that the 
square root of that number is to be taken, or employed ; as 
V64 = 8. 

^ Sign of the cube root; and indicates that the cube root of the num- 
ber to which it is prefixed is to be employed, instead of the num- 
ber itself; as ^^^64 = 4. 

* To be squared, or the square of; shows that the square of the number 

to which it is affixed is the quantity to be employed ; as 12*' -r- 
6 = 24 ; that is, that, the square of 12, or 144 -7- 6 = 24. 
^ Indicates that the cube of the number to which it is subjoined is to 
to be used ; as 4^ = 64. 

• Decimal point, or separatrix. See Decimal Fractions. 

Vinculum. Signifies that the two or more quantities over 

which it is drawn, are to be taken collectively, or as forming^ 
one quantity ; thus, 4 -|- ^ X 4 = 40 ; whereas, without the 
vin culum, 4 + 6 X 4 = 28 ; also, 12 — 2X3 + 4 = 2 ; and 
V5^Zr32 = 4. So, also, V (5^ — 32 ) = 4, and (4 + 6) X 4 
= 40. 



42 ( half of 42 or ) 

2 ( half of the square of 4 ) ~~ 

(42 \2 
— I (the square of half the square of 4) = 64. 



452 or 4(5)2 (half the square of 5.) 
(46)2 (the square of half 6) 
(25)2 (the square of twice b,) 



FOREIGN MONEYS OF ACCOUNT: 

THEIR DENOMINATIONS, RELATITE VALUES, AND VALUES IN FEDERAL 

MONEY, 

The value in Federal money, affixed to any particular denomina- 
tion of a Foreign money of account, in the following tables, is the 
intrinsic par thereof, as near as practicable. It is based upon the 
standard weight and purity of the coins coined especially to repre- 
sent that denomination, compared with the standard weight and 
purity of the coins of the United States that represent the dollar^ 
and is the United States Customs value of that denomination for 
computing duties. The denomination itself, to which the Federal 
value is immediately affixed, is usually the unit or ultimate money 
of account of the country especially referred to. It is a money of 
account in that country always ; but not always in that country 
the name of a national coin. It Ls not always, even, repres^ted by 
a single national coin. Thus, in Great Britain, until comparatively 
recently, there was no single British coin of the value of one pound. 
That denomination is now represented by a single gold coin, called 
a sovereign. 

Foreign. U. States, 

ALGIERS. — Algiers, Bona, &c. : 100 centimes = 1 Li- 

vre, = $0,187 

29 aspers = l tomin, 8tomins = l pataka-chicas, 

3 patakas-chicas= 1 Piastre, - - = 0.21 

ARABIA. —Mocha, Jidda : 80 caveers = 1 Piastre, - = 0.823 

1 wakega ybr gold and silver = 4:S0 troy grains. 
AUSTRIxV. — 100 centesimi = 1 Lira Austriache, - = 0.1622 

Vienna, Trieste, Prague (Commercial) : 4 pfennige 
= 1 kreuzer, 60 k, = 1 Gulden or Florin Austri- 
ache = 21J of a Cologne mark of fine silver = 
ISOtV^ "troy grains, - - - -= 0.4858 

. 6 hellers = 1 groschen, 80 g. = 1 Florin, - = 0.4858 

li groschen = 1 kreuzer. 

14 florins = 1 current Thaler. 

2 florins = 1 Specie Thaler. 

1 florin Austriache = 5xV lire de Trieste = 5^ lire 

di piazza. 
1 mark Austrian = 4332^ troy grains. 



2 a FOREIGN MONEYS OF ACCOUNT. 

Foreign. U. States. 

AZORE ISLANDS. — Corvo, Fayal, Flores, Graciosa, 
Pico, Terceira, St. Michael, St. Mary's: 1000 
reas = 1 Milrea, - - - - = $0.83i 

BALEARIC ISLANDS. — Majorca l.—Pahna: Mi- 
norca I. — Port Mahon : Siune as Cadiz, Spain. 
BELGIUM. — Antwerp, Brussels, Liege, Mechlin, 
Ghent, &c. : 
10 centimes = 1 decime, 10 d. = 1 Franc, - = 0.187 
24 mitres or 8 Brabant penning or G ILarde or 2 
groote = 1 stuiver, stuivers = 1 schclling, 20 s. 
= 1 Pond Flemish, - - • - - = * 2.377 

G gulden or 2i diuilder = 1 Pond Flemish. 
BERMUDAS I. —4 farthings = 1 penny, 12 pence = 1 

shillhig, 20 sliillings = 1 Pound, - - = 3.00 

BOURBQfJ ISLANDS. — 100 centimes = 1 Franc, -= 0.187 
BR^VZIL. — Ararat i, Bahia, Maranhain, Para^ Pcmam- 
' huco, Portahgra, Rio Janriro, <fcc. : 1000 reus = 

1 Milrea, = 1.042 

BONAIRE I. — 48 stuivers or 8 redls = 1 Piastre, = 0.73 

Central and South Arnica. 

Balize, Campeche, Guatimala, Honduras, Laguna, Leon^ 

Nicaragua, San Juan, San Salvador, Sisal, &c. : 
Buenos Ayrcs, Callao, Carihagma, Coipninbo, (ivtiyaqml, 
Lag u ay r a, Lima, Maracayho, Montevideo, Rio 
Hacha, Tnnillo, Valparaiso, &c. : 
2 ciiartilli = 1 medio, 2 inedios = 1 Real, - = 0.124 
8 reiile = 1 Peso, - - - - -= 1.00 

Pound of Honduras, - - - _ = 3.00 

Berbicc, Dcnurara^ Kssn/uiho, Surinam : 

8 duyt «= 1 stuiver, 2() s. = 1 Florin or (iulden, = 0.33^ 
Cayenne: 100 centimes = 1 Franc or Livre, - - = 0.187 

CANARY ISLANDS. — Grand Canary, Tcncriffe, 
Pahna, &c. : 
34 maravedi = 1 Real (current), - - = 0.074 

CANDIA I. — 80 aspors or 33 mcdini = 1 Piastre, = 0.018 
CAPE VERD I. — 1000 reas = 1 Milrea, - - = 

CHILI. — Coquinibo, San Carlos, Santiago, Valdivia, 
Valparaiso, &c. : 
4 cuartilli = 1 medio, 2 m. = 1 real, S r. = 1 Peso, = 1.00 
CHINA. — Amoy, Canton, Macao, Nankin, Pekin, 
Shanghac, &c. : 
10 cash = 1 candarine, 10 candarines = 1 mace, 10 

mace = 1 Tael, - - - - = 1.48 

Hong Kong I. — Same as Great Britain. 



FOBEIGN MONEYS OF ACCOUNT, ^ a 3 

Fordgn, U. States, 

CORSICA I. — 100 centimes = 1 Franc, - - = $0,187 

CYPRESS I. —Same as Constantinople (Turkey), 
DENMARK. — Copenhagen, &c. : 12 pfennige = 1 skil- 
ling, 16 s. =1 mark, 6 m. = 1 Rjksbankdaler = 
■^Y of a Cologne marc, or 194.98 troy grains of 
pure silver, - - - - - = 0.5252 

2 Ryksbankdalers = 1 Speciesdaler, - - =1.05 

EGYPT. — Alexandria and the Delta : 8 borbi or 6 fiorli 

or 3 aspers = 1 medimne, 40 m. = 1 Piastre, = 0.048 
20 piastres = 1 redl (a gold coin), - - - = 0.968 

Cairo ; 80 aspers or 33 medimni = 1 Piastre, - = 0.048 
FRANCE. — Standard for gold and silver coins = ^^ 
fine, each : Relative values — gold to silver as 15 
to 1. 
10 centimes = 1 decime, 10 d. = 1 Franc = -^(j^ 

kilogrammes of fine silver = 69.449 troy grains, = 0.18706 
12 deniers = 1 sou, 20 s. = 1 Livre tournois (old) 
= 1^ franc. 
GERMANY. — The mark of Cologne, divided into 8 unze = 64 
quent = 256 pfennig = 512 heller =-4352 eschen = 65536 richt- 
pfennig, is employed at the mints, for weighing gold and silver 
coins, throughout Germany ; this mark = 3607i troy grains. 

The standard for the Ducat, at present, throughout Germany, ia 
72- or 23 1 carats fine, its weight, ^y of a Cologne mark. This piece, 
therefore, should weigh 53.84 troy grains, and contain 53.09 troy 
grains of fine gold = §2.286. 

The standard for the Pistole or Zehn Gulden piece, of the Con- 
vention of 1753, is f I" or 21} carats fine ; its weight, ^3- of a Cologne 
mark. This piece, therefore, should weigh 103.06 troy grains, and 
contain 93.4 troy grains of fine gold = §4.022. 

The standard for the Specie Thaler, or Rixdollar afiective, of the 
Convention of 1753, is f or 13 loth 6 gran fine, its weight ^\ Co- 
logne mark = $0.9716 ; and the species Florin is 4 thereof = 
$0.4858. 

The standard value of the Current Thaler or RixdoUar of account, 
established about 1775, is | RixdoUar affective = 1^ specie Florins 
= $0.7287. 

The Thaler of the Convention of 1838 contains y'^- Cologne mark 
of fine silver = §0.694 ; and the Gulden or Florin of that Conven- 
tion = f thereof = §0.3966 : 1| gulden- = 1 Thaler. 

The standard of purity for the coins established by the Convention 
of 1838, are : Two Thaler piece = -^^ ; thaler, | thaler, J thaler, 
each, = I ; florin, 4 florin, each = ^tj> less fractions = J. 



4 a . FOREIGN MONinrS OF ACCOUNT, 

Three Thalers of the Convention of 1838, by the eptimate of rela- 
tive values then established (gold to silver as 14,56 to 1), = 1 Ducat ; 
while five of the current RixdoUarSy established in 1775, by the then 
existing estimate of relative values (gold to silver as 14.483 to 1), 
^= 1 Pistole, or 10 guilder piece, above referred to. 

Most of the moneys of account throughout Germany, including 
Austria and Prussia, are based upon some portion of the fore- 
going. 

Foreign, U. States, 

Bremen : 5 schwaren = 1 groot, 72 g. or 2 florins = 1 

RixdoUar = -J standard Carl d'or, - - = $0w8| 

Frankfort : 4 pfennig = 1 kreuzer, 60 k. = 1 Gulden = 

y\y Convention pistole, - - - - = 0.4022 

90 kreuzer = 1 Rixdollar of account, - = 0.6033 

14 albus = 1 kraiser-groschen, 30 k. = 1 Rixdollar, = 0.6033 
4 kreuzer = 1 batzen, 22^ b. = 1 Rixdollar, - = 0.6033 

120 kreuzer or 2 gulden or Ij Rixdollar = 1 Species 
Rixdollar, - - - - - = 0.8<Vi4 

Hamburg, Luhec : 12 pfennig = 1 schilling, 16 Bchillinge 
= 1 mark. 

1 mark current = $0.28. 1 mark banco, - = 0.35 
3 marks banco = 1 Specie Rixdollar, - - = 1.05 

Baden. — Carlsruhe, Heidelberg, &c. : Same as Bavaria. 

Bavaria. — Augsburg, Bamberg, Bayrcuth, Munich, 
Nuremberg, Ratisbon, Wurtzbicrg, &c. : 4 denari 
= 1 kreuzer, 60 k. = 1 Florin, - - - = 0.4858 

2 kreuzer = 1 albus, 2 a. = 1 Ixitzen, 15 batzens 
= 1 Florin. 

14 florins = 1 Rixdollar current, - - = 0.7287 

2 florins = 1 Specie Rixdollar. 

Hanover. — Emden, Gottcngcn, Hanm^er, Osnahnrg, <fec. : 

12 pfennig = 1 gute groschen, 24 g. = 1 Rixdollar, = 0.7287 
IJ Rixdollar = 1 Specie Rixdollar. 

60 kreuzer = 1 gulden, IJ g. = 1 Thaler, - = 0.694 
Hesse. — 12 hellers or 9 pfennig = 1 albus, 32 albus 

= 1 Rixdollar, - - - - - = 0.7287 

24 marien groschen = 1 reichsflorin, I4 r. = 1 Rix- 
dollar. 
60 kreuzer = 1 gulden, 1| g. = 1 Thaler, - = 0.6(M 
HoLSTEiN, Altona, Kiel, &c. : 12 pfennig = 1 schilling 

lubs, 16 schillinge {of Lubec) = 1 mark, - - = 0.35 

3 marks = 1 Speciesdaler of Denmark, 
Mecklenburg. — Rostock, Wismar, &c. : Same as Hano- 
ver. 

Olbenburg. — Same as Bremen; also, same as Ham- 
burg. 



FOBEIGN MONEYS OF ACCOUNT. oS 

Foreign. U. States.^ 

Baxot^y.^^ Dresden, Leipsic, &c. : 12 pfennig == 1 gute 

groschen, 24 g. = 1 Species Thaler, - - = $0.7287 

16 gute groschen= 1 reichflorin, 2 r. = 1 Specie 
Rixdollar. 
Saxe. — Gotha, Weimar : Same as Haxoyer, 
Saxe generally and Nassau : 4 hellers = 1 kreuzer, 60 

kreuzers = 1 Gulden, - - - - = 0.4022 

IJ gulden = 1 Rixdollar current, 
WuRTEMBURG. — Halle, Stuttgord, Vim, &c. : Same as 

Saxe and Nassau. 
GREAT BRITAIN. — ^er/my money: Standard for 
silver coins = fj fine ; for gold coins = yj- fine. 
Relative values, gold to silver as 14.288 to 1. 
4 farthings = 1 penny, 12 p. = 1 shilling, 20 s. = 

1 Pound = $4,866.* U. S. Customs value, - = 4.84 
GREECE. — 12 denari = l soldo, 20 s. = 1 Lira or 

Dra-chma, - - - - - = 0.163 

HOLLAND. — Standard for silver coins = ynVry fine. 
Standard for gold coins — Gouden Willem (10 
florins) fractions and multiples = -ijy^y^ fine — 
Ducat and multiples = xV^rd fine ; weight of 
Ducat = 3.494 grammes = §2.2827. Relative 
values — gold to silver as 15.004 to 1. 
100 centimes = 1 Florin or Gulden = 9.45 grammes 

of fine silver = 145.843 troy grains, - - = 0.^928 

24 florins = 1 Ryksdaalder, - - - = 0.982 

10 florins in gold = 6.056 grammes fine gold = 
$4.0251, Custom-House value of Florm = $0.40. 

India and Malaysia or East Indies. 

British Possessions. — Standard of purity for gold 
and silver coins = \h fine, each, 
Hindostan. — Bombay^ Siirat, Tatta : 100 reas = 1 quar- 
ter, 4 q. = 1 Rupee = 165 troy grains of fine 
silver, - - - - - - = 0.4444 

* For all time since 1816, the government of Great Britain has estimated gold and silver 
as 14.2879 to 1. The ;)oi/nrf sterling of mint silver weighs 1745. 4:54 grains, and contains 
1614.545 grains of fine silver. The value of the pound sterling of silver, therefore, rated 
by the United States standard of o71i grains of fine silver to the dollar^ is $4,349. The 
value of the silver shilling, of full weight, is $0.2174. The pound sterling of mint gold 
weighs 123.274 grains, and contains 113.001 grams of fine gold. The value of the pound 
sterling of gold, rated by the United States standard of 23.22 grains of fine gold to the 
dollar^ is $4,866. At the old intrinsic par between the two currencies, viz., 4 shillings 
and Q pence sterling to the dollar, or $4.44^- to the £, the par of exchange is Q^per 
cent. Gold is the money standard in Great Britain, and the silver coins of that country 
are not legal tender at home in sums exceeding £2. 



&a FcmmGN monIts of account. 

Foreign, U. States. 

Calcutta: 12 pice = 1 anna, 16 annas = 1 Rupee, = $0.4444 
1 Arcot rupee = 1 Sicca rupee = $0.442^tf . 
1 Mohur or Gold Rupee == 165 troy grains fine gold 

= $7.1033. 
A Lac of rupees = 100.000 rupees. 
A Crore of rupees = 100 lacs of rupees. 
Madras. — 12 pice = 1 anna, 16 annas = 1 Rupee, = 0.4444 
80 cash = 1 fanam, 42 fanams = 1 Pagoda, - = 1.851 
34 old Sicca rupees = 1 Pagoda. 
1 Sicca rupee less 16 per cent. = 1 current rupee. 
Cochin: 12 pice = 1 anna, 16 annas = 1 Rupee, =» 0.4444 

4 fknams = 1 schilling, 5 schillings = 1 Rupee. 
Goa : 18 budgerooks = 1 vintim, 4 v. = 1 tuiiga, 5 t. 

= 1 Pardo, - - - - - = 0.90 

Pondicherry : 60 cash = 1 fanam, 24 f . = 1 Pagoda, «= 1.84 
Banca I. — Same as Ratavia {Java L) 
Borneo I. — Borneo, Pcntaniah, SojnboSj &c. : Same as 

Rata VIA (Java I.) 
Celebes I. — Macassar: Same as Batavia (Java I.) 
Ceylon I. — Colombo : 4 pice = 1 fanam, 12 fanams = 

1 Ryksdaalder, = 0.3928 

Java I. — Batavia, Samarang, Sarakarta: 5 duyt = 1 
stuiver, 2 stuivers = 1 duljhcl, 3 du])l>elo = 1 
schilling, 4 schillingo = I Florin, - - = 0.3928 

U. S. Customs value of tlic Florin = $0.40. 
Ma'lacca. — Malacra : 4 duyt = 1 stuivi^r, 6 stuivers = 

1 schilling, 8 s<'hillingc = 1 Ryksdaalder, - = 0.795 
Molucca Islands or Spice I. — Same as Batavia, Java I, 
Philippine I. — Luzon 1. — Manilla, &c. : 34 mara- 

vedi = 1 rcill, 8 reals = 1 Peso, - - = 1.00 

SiAM. — Bangkok, &c. : 4 tical or bate = 1 tael, 100 t. 

= 1 Pecul, - - - - - = 0.617 

Singapore I. — Same as Mai^\cca. 
SooLoo Islands. — Same as China. 
Sumatra I. — Achcen: 4 copangs = 1 mace, 4 m. = 1 

pardo, 4 p. = 1 Tael, - - - - = 4.16 

Bencoolen : 8 sa tellers = 1 soocoo, 4 s. = 1 Redl, = 1.10 
IONIAN ISLANDS. — Cephalonia, Corfu, Ithaca, 
Paxos, Zante, &c. : 12 denari = 1 soldo, 20 s. = 
1 Lira, - - - - - = 

ITALY. — 100 ccntesimi = 1 Lira Italiani, - - = 0.187 

Ecclesiastical States. — Standard for silver coins = 
f 11 fine ; for gold coins = TaVa fiiie. Relative 
values, gold to silver as 15.526 to 1. 



FOREIGN MONEYS OF ACCOUNT. a 7 

Foreign, U. States, 

Ancona: 12 denari = 1 soldo, 20 s. = 1 Scudo, - = $1,007 

100 bajochi or 80 bologni or 10 paoli = 1 Scudo. 
Bologna^ Ferrara : 12 denari = 1 soldo, 20 s. = 1 Lira, = 0.201 

5 lire or 10 paoli or 100 baiochi or 500 quattrini = 

1 Scudo, = 1.007 

Rome : 5 quattrini = 1 baiocho, 10 b. = 1 paolO, 10 p. 

= 1 Scudo = y^- libbra (373.78 troy grains) of 

fine silver, - - - - - = 1.007 

Sequin = -f f f fine = ilfjj libbra of fine gold. 
Lucca. — 12 denari = 1 soldo, 20 s. = 1 Lira, - = 0.243 

12 denari = 1 soldo, 20 s. = 1 Scudo correnta, = 1.007 

1 scudo d'oro of Lucca = $1,127. 

12 denari di cambia = 1 soldo di cambia, 20 s. = 1 

Scudo di cambia, - - - - = 1.068 

Lombard Y and Venice. — Standard for silver coins = J§ 

fine ; for gold coins = |§ fine. Relative values, 

gold to silver as 16 to 1. 

Bergamo, Mantua, Milan, Padua, Verona, Venice, &c. : 

100 centesimi = 1 Lira Italiani, - - = 0.187 

100 centesimi = 1 Lira Austriache, - - = 0.162 

12 denari = 1 soldo, 20 s. = 1 Lira correnta di Mi- 
lan = 5^Jf (J marcs of fine silver = 52.531 troy 
grains, - - - - - = 0.1415 

1 Lira picola di Bergamo, di Verona or di Venice, = 0.098 

1 Lira di Mantua, - - - - - = 0.058 

MoDENA, Parma. — 100 centesimi = 1 Lira, - = 0.187 

Naples. — Naples, Salerno, &c. : 7200 accini = 360 
trapesi = 12 oncie = 1 marco or libbra = 4950.53 
troy grains. 

Standard for gold and silver coins = yVcnr fii^e, each. 

Relative values, gold to silver as 15.373 — to 1. 

10 grani = 1 carlino, 10 c. = 1 Ducato = ^^7 Nea- 
politan libbra of fine silver = 295.55-|- troy 
grains, - - - - - = 0.796 

li ducati = 1 scudo or crown. 
Sardinia : 4608 grani of 24 granottini, each = 1152 
carati = 192 denari = 8 oncie = 1 marco = 3795 
troy grains. 

Standard for gold and silver coins = || fine, each. 

Relative values, gold to silver as 15.545-[- to 1. 

100 centesimi = 1 Lira Italiani, - _ =_ 0,187 

12 denari = 1 soldo, 20 s. = 1 Lira di Sardinia = 

2 V i^^i'co of fine silver = 130.86 troy grains, -= 0.3525 
Genoa : 5| lire fuori banco = 1 Pezza, - - = 0.8854 



8 a FOREiaN MONEYS OF ACCOUNT. 

Foreign, U. Slates » 

4f lire di cambio = 1 scudo of Exchange, - ' - = $0.7187 

lOxVis" lire moneta buona = 1 Scudo d'oro, - = 1.6GG3 

Tuscany. — Florence — moneta buona : 12 denari = 1 

soldo, 20 soldi = 1 Lira, - - - = 0.1566 

12 denari di ducato = l soldo di ducato, 20 s. = 

1 Ducato or scudo correnta, - - - = 1.0962 

7 lire = 1 ducato. 7i lire = 1 scudo d'oro. 
Leghorn. — moneta lun^a : 12 denari di lira= 1 soldo 

di lira, 20 soldi dilira = 1 Lira, - - = 0.15 

12 denari di pezza = 1 soldo di pezza, 20 soldi di 

pezza = 1 Pezza,- - - - -= 0.9004 

G lira = 1 pezza. 

7 lir(3 moneta l)uona = 1 Scudo corrente, - = 1.09G2 
74 lire monetii buona = 1 Scudo d'oro, - - = 1.1745 

JAPAN. — Malsmai/, Miaco, NangaMiki, Osaca^ Ycdo, &c. : 
10 cash = 1 candarine, 10 c. = 1 mace, 10 m. = 1 

Tacl, == 1.4074 

MADEIRA ISLANDS.— 1000 reas = 1 Milrca, - = 1 .00 

MALTA 1. — 20 grani = 1 taro, 12 t. = Scudo, - = 0.406 
G piccioli = 1 carlino, 2 c. = 1 tiiro, 12 t. = 1 

Scudo, - - - - - - = 0.406 

2.i Hcudi = 1 Pezza, - - - - = 1.015 

MAUKITIUS I. — Port Louis, &c. : 100 cents = 1 Dol- 
lar, = 0.0853 

20 sols = 1 Livre colonial = <| Franc of France, = 0.09353 
10 livres = 1 Dollar. 
M EX ICC). — A rap u ho , Tamjuco , Vera Cruz , &c . : 

G grani = 1 cuarto, 2 c. = 1 medio, 2 m. = 1 Roiil, = 0.125 

8 reals = 1 Peso, - - - - = 1.00 
MOROCCO. — /V^, Mogadorc, &c. : 24 fluce = 1 blan- 
ked, 4 b. = 1 once, 10 onces = 1 Mitkul, = 0.7407 

NORWAY, —^er^rn; IG skillinge = 1 mark, G m. = 

IRyksdaler, = 0.5252 

2 Ryksdalers = 1 Speciesdaler. 

Christiania, &c. : 12 })fenni<je=l skilling, G s. = 1 
ort, 4 orte = 1 Ryksdaler = 1 mark banco of 
Hamlnirg, - - - - - = 0.3501 

3 Rykvsdaler = 1 Speciesdaler. 

PERSIA. — Bushirc, &c. : 5 denari = 1 kasbeque, 10 k. 

= 1 shafrcc, 2 s. = 1 mamoode, 2 m. = 1 abasse, 

50 a. = 1 Toman, - - - - ■= 2.233 

2 kas])equi = 1 denaro-biste. 
PORTU (J AL : Standard for silver coins = f || fine — for 

gold coins = { j fine. Relative values, gold to 

silver as 15.356^- to 1. 



FOEEIGN MONEYS OP ACCOUNT. a 9 

Foreign. U. States, 

Method of writing and reading quantities : Ejc. — 
rs. 5 : 600 750 = 5,600 milreas and 750 reas. 
1000 reas = 1 Milrea = the silver coroa = ff § 

marco of fine silver = 415.435 troy grains, - =s $1,119 
1000 reas = 1 Milrea current — fluctuating, about 

= $0.96. 
1 Milrea, paper — fluctuating, about = $0.81. 
480 reas = 1 Crusado. 
PRUSSIA. — Standards and relative values, same as 
given under Gerjiany. 
Berlin, Brandenburg , Dantzic, Potsdam, Magdeburg, 
Stetin, &c. : 
12 pfennig = 1 gute groschen, 24 g. = 1 Rixdollar 

or Thaler current = 1^ V specie thaler, - = 0.7287 

li Rixdollar = 1 Thaler banco = $0.91/^-. 
li florins = 1 Thaler specie, - - - =» 0.694 

Cologne: 12 hellers = 1 albus, 80 a. = 1 Rixdollar 

= -^jj Convention pistole = IJ florins d'or, - = 0.6033 
78 albus = 1 Rixdollar current. 
120 fettmangen = 90 kreuzer = 30 groschen = 20 
blafferts = 3i Cologne florins = 2 heron florins 
= 14 rader florins. 
Aix la Chapelle, Crevelt, Elbcrfeldt, &c. : 

4 pfennig = 1 kreuzer, 60 k. = 1 florin, IJ f. = 1 

Thaler, --.--= 0.694 

Brunswick: 8 pfennig = 1 marien-groschen, 36 m. 

= 1 Thaler, = 0.694 

Konigsberg : 6 pfennig = 1 schilling, 3 s. = 1 gros- 
chen, 30 groschen = 1 florin, 3 f . = 1 Thaler, = 0.694 
8 specie gute groschen of Berlin = 30 groschen of 
Konigsberg. 
PRINCE OF WALES I. — 10 pice = 1 copang, 10 c. 

= 1 Dollar, = 1.00 

PROVINCES OF New Brunswick, Nova Scotia, New- 
foundland, AND THE CaNADAS I 
4 farthings = 1 penny, 12 p. = 1 shilling, 20 s. = 

1 Pound, -----= 4.00. 
RtlSSIA. — Standard for silver coins = f fine, for gold 
coins = yg- fine. Relative values, gold to silver 
as 15^^^ to 1. 
Archangel, Cronstadt, Hehingfors, Odessa, Revel, Se- 
vastopol, St. Petersburg, &c. : 
10 kopecs = l^grieven, 10 g. = 1 Rublyu (ruble) 

= ^\ funt of fine silver = 278.47 troy grains, = 0.75 



10 a 



FOREIGN MONEYS OP ACCOUNT. 



Foreign. 
2 denushkas or 4 polushkas = 1 kopec. 33J altius 
= 1 ruble. 
Riga. — Same as St. Petersburg, also — 

30 groschen = 1 florin, 3 f . = 1 Rixdollar. 1 Alber- 
tus dollar, _ - - - - 

SARDINIA I. — 100 centesimi = 1 Lira Italiani, 

1 Lira di Sardinia, - - _ - 

SICILY I. — Standards of purity and relative values, 
same as Naples. 
6 picioli = 1 grano, 20 g. = 1 taro, 30 t. = 1 Oncia 

= ef ^^^'^^P^l^^^^^^ 1^^^^^^^ ^^f^rie silver, 
8 picioli = 1 punti, 15 p. = 1 taro, 10 t. = 1 ducato. 
G tari = 1 fiurino or flurin, 2 f . = 1 scudo, 2i s. = 

1 oncia. 

SPAIN. — Standard for silver coins since 1786, peso and 
J peso = xff line ; peseta, real and ^ real = f | 
line ; fur guld coins = J line, except the curonilla 
{gold dollar) = } |§ fine. Relative values, since 
1780, gold to silver as 10.39 to 1. 
Real vellon = 2V P^so duro, - - . 

Real de platii nuevo = j'^^ peso duro, - 
Real de plata Mexicana = i peso duro, 
Redl de plata antifjuas = :^2^ pe«o duro, 
Rcill d'Alicant = jVj^ peso duro, - - - 

Redl do Valencia = j^/j peso duro, 
Rciil curmntc de Gi))raltur = y^ peso duro. 
Real de Catiilonia = ^Vs V^^^ duro, - 
Reiil ardita de Catalonia = -^^^ peso duro, 
8 reiile de plata antiipias = 1 Piastre or peso of ex- 
change = %^ peso duro, - - - 
Peso duro = x/j marco of fine silver = 371.9 troy 
grains, ------ 

40 dineri = 10 comadi = 8 blanci = 4 maravedi = 

2 ochavi = 1 quarto, 4i quarti = 1 sualdo, 2 s. 
= 1 Real. 

Alicant : 34 maravedi = 1 re:il, 10 r. = 1 Pia.stre, = 
Barcelona, Tortosa : 2 malli = 1 dinero, 12 d. = 1 
sualdo, 20 s. = 1 Libra = 10 redle ardita de 
Catalan, - - - - - : 

Bilboa, Carthagcna, Madrid, Malaga, Santander, 
Toledo: 
34 maravedi = 1 real, 15 r. = 1 Peso scn^lo. 



U. States. 



= $1.00 
= 0.187 
= 0.354 



= 2.388 



= 0.05 

= 0.10 

= 0.125 

= 0.0943 

= 0.0754 

= 0.0566 

= 0.0835 

= 0.0808 

= 0.0539 

= 0.7543 

= 1.0018 



0.7543 
0.5388 



FOREIGN MONEYS OF ACCOUNT. all 

Foreign. U. States. 

15xV ^edle = 1 peso de plata or Piastre, - = $0.7543 

4 piastres = 1 doubloon de plata or pistole of ex- 
change. 
Cadiz, Sevilla: 34 maravedi or IG quarti = 1 Redl, = 0.0943 

8 reale = 1 Piastre, 10| reale = 1 Peso diiro. 
Gibraltar: 34 maravedi = 1 real, 9 r. = 1 Piastre, = 0.7543 

12 reale = 1 Peso duro. 
Valencia: 12 dineri = 1 sualdo, 2 s. = 1 real, 10 r. 

= 1 Libra, - - - - - = 0.5657 

Ij libra = 1 Piastre or peso of account, - - = 0.7543 

24 dineri = 1 real, 10 r. = 1 Peso duro, - = 1.0018 

SWEDEN. — Standard for gold coins (ducats, multiples 
and fractions) = ^ IJ ^^^ '•> ^'^^ silver coins = fy 
fine. Kelative values, g(jld to silver as 14.092 to 1. 
Carlscrona, Gcjle, Goitenburr/, Stockhalm, &c. : 

12 rundstycken or ore = 1 skiliing, 48 s. = 1 Riks- 
• daler = ^\ mark of fine silver = 393.68 troy 
grains, - - - - - -= 1.0604 

100 centimes or skillings = 1 Riksdaler, - = 1.0004 

SWITZERLAND. — 1 Li\Te de Suisse, of the convention 
of 1814, = 14 lii'}'cs tovrnois of France = 27 
francs oi'Fvancc J - - - -= 0.2771 

Berne, Basle, Lausanne, Lucerne, Pay de Vaud : 
10 rappen = 1 batz, 10 batzen = 1 Livre de Suisse. 
12 deniers = 1 sou, 20 sols de Sui.ssc = 1 Livre. 
10 rappen = 1 batz, 15 b. = 1 Florin or Guilder, = 0.41 50 
8 hellers = 1 kreuzer, GO k. = 1 Florin. 
Geneva : 12 deniers = 1 sou, 20 s. = 1 Livre = 3^ 

florins petite monnie = 1^ francs of France, = 0.3117 

3 livres = 1 Ecu or Patagon, - - = 0.8313 

Neufchafcl: 100 rappen = 1 Franc or Livre de Suisse. 
12 deniers =1 sol, 20 s. = 1 Livre tournois de 

Neufchatel = 2^ livers foible, - - - = 0.2628 

St, Gaul: 480 heller = 240 pfennig = GO kreuzer = 
15 batzen = 10 skiliing = 1 Florin or Guilder. 
1 florin current = 2^ francs of France, - = 0.4365 

1 florin specie, - - - - -= 0.5187 

Zurich : GO kreuzer of 8 hellers each, or IG batzen of 
10 augsters each, or 40 skillings of 12 hellers each 
= 1 Guilder or Florin, - - - = 0.4365 

TRIPOLI. — 100 paras = 1 Piastre or Ghersch, ghersch 

of 1832, = 0.10 

TUNIS. — Tu7iis, Biserta, Susa, &c. : 2 burbine = 1 



12 a FOREIQN MONEYS OF ACCOUNT. 

Foreign. U, Slates. 

asper, 52 a. or 16 carobas = 1 Piastre,* piastre 

of 1838, =$0,128 

TURKEY. — Constantinople : 3 aspers = 1 para, 40 p. 
= 1 Piastre or Ghersch.* 
With the Dutch, French and Venetians, 100 aspers = 1 

Piastre. 
With the English and Swedes, 80 aspers = 1 Piastre. 
500 piastres = 1 chise ; 30,000 piastres = 1 kitz ; 
100,000 piastres = 1 juck. 
Smyrna : 40 paras or medini = 1 Piastre or Gooroosh. 
12 tomans = 1 Piastre or Gooroosh. 

West Indies, 

Cuba I. — Cardenas, Cienfuecjos, Havana, Matanzas, 
Mariel, Nuc vitas, Porto Principe, Sagua la Grande, 
St. Jago, &c. : 
34 maravedi = 1 redl, 8 r. = 1 Peso, - -=1.00 

Hayti I. — Aux Cayes, Cape Haytien, Port au Prince, 
San Domingo, &c. : 
100 centesiini = 1 Dollar or Peso duro = 11 escu- 

lini, .--.-= 1.00 

1 dollar Ilaytien currency = 7i csculini, - - = 0.G6 

Porto Rico I. — Guayama, Mayaguez, St. Johns, 
Ponce, ttc. : 
34 maravedi = 1 redl, 8 r. =: 1 Peso, - - =1.00 

British Islands. — Anguilla, Antigua, Barbuda, Do- 
minica, Grenada, Montscrrat, Nevis, St. Kills, St. 
Lucia, St. Vincent, Tobago, Tor tola, Trinidad, 
Virgin Gorda : 
4 farthings = 1 penny, 12 p. = 1 shilling, 20 s. *= 

1 pound, - - - - - = 2.222 

Nassau and the Bahamas generally : 

4 farthings = 1 penny, 12 p. = 1 shilling, 20 s. = 

1 Pound, = 2.4S5 

• Pound of Turks I. - - - - -= 3.00 

Barbadoes I. — Bridgetown, Sea. : 1 Pound, - = 3.20 

Jamaica I. — Falmouth, Kingston, Morant Bay, Savan- 
nah la Mar, &c. : 
4 farthings = 1 penny, 12 p. = 1 shilling, 20 s. =: 

1 Pound, - - - - - -=3.00 

* The coins of the Turkish government, owing to frequent and oft-repeated deterioration 
by enactments, have no definable standard value whatever. Bills of exchange on Turkey 
are usually drawn in Spanish dollars. The value of the silver piastre of Turkey, of full 
weight, of 1775, is $0,446 } of that minted in Tunis in 1787, $0,259 •, of that of Turkey of 
1818, $0,182, and of that of 1836, $0,128, while that issued only a few years since, is 
worth, mtriusically, but about 4 cents. 



FOREIGN MONEYS OF ACCOUNT. a 13 

Foreign. U. States, 

Danish Islands. — Santa Cruz, St. John, St. Thomas, 
St. Bartholomew : 
12 skillings = 1 bit, 8 b. = 1 Ryksdaler, - = SO. 64 
100 cents = 1 Ryksdaler. 
12i bits = 1 Spanish dollar. 
Dutch Islands. — Saba, St. Eustatius, St. Martin: 
6 stuivers = 1 redl, 8 r. = 1 Piastre, - - = 0.73 

11 reals or Esculins = 1 Spanish dollar. 
French Islands. — Deseada, Guacleloupe, Mariega- 

lante, Martinique : 

12 deniers = 1 sol, 20 s. = 1 Livre = | livre tour- 

nois, ------= 0.1232 

4 farthings = 1 penny, 12 p. = 1 shilling, 20 s. = 

1 Pound, - - ^ - ^ - - = 2.222 

Little Antilles, generally, Same as Mexico. 

B 



14 a 



FOREIGN LINEAR AND SURFACE MEASURES 
REDUCED TO UNITED STATES. 

Foreign. 

ABYSSINIA. — Massuah : 8 robi= 1 derah or pic, = 

ALGIERS. — 10 decimetres = 1 metre, - - = 

8 robi=l pic. Pic, Moorish, for linens, - = 

Pic, Turkisjt, for silks, &c., - = 

ARABIA. — 1 kas8aba= 12.31 ft. Mile, 

Aden : 8 robi= 1 yard or pic, - - - = 

Jidda: 8robi=lpic, - - - = 

Mocha: 8 gheria=l covid. Covid {land), - = 

Covid (/or 2>o;i, cfc), - - = 

8 robi= 1 ^Qz, _ - . = 
AUSTRIA. — {Imperial, or legal and general) : 
Vienna, Trieste, Prague, Lintz, cj-c. ; 

12zoll = lfus, - - - - = 
29^zoll=lelle, 

6 fiis= 1 klafter, 4000 k. = 1 meile, 

10 fu8= 1 ruth (^i7</er5'), - - - = 

3 metzen = 1 joch, - - - = 
{Special and local) — 

Upper Austria. — Lintz, <5fc. : 1 ellc, - - = 

BouEMiA. — Prague, djc. : 2 fus — 1 clle, - = 

4 ellc = 1 duniplachter, - - - = 
Hungary. — 1 fus = 1.037 feet. 1 ellc, 
Moravia. — 2ffus=lelle, - - - = 

AZORE ISLANDS. —Same as Lisbon {Portugal). 
BALEARIC ISLANDS. — 3 pie or 4 palma = 1 
vara, 2 vara = 1 cana. 
Majorca. — 1 cana, - - = 

Minorca. — 1 cana, - - - - = 

BELGIUM. — 10 strecp = 1 duim, 10 d. = 1 palm, 
10 p. = 1 el. = 1 metre of France, - = 

10 el= 1 roed, 100 r. = l mijl, 
2 J fus = 1 aune. 
Antwerp : Anne for cloths, - - - = 

Anne for silks, - - - = 

J5ru55eZ5; 1 aune = 0.761 yards. Vaem, - = 



u. » 


Stales 


0.682 
1.094 


yard 


0.519 


(( 


0.692 


(( 


1.22 


miles 


0.95 
0.743 


yard 


1.58 


feet 


2.25 


(t 


0.694 


yard 


1.037 


feet. 


0.852 
4.712 


yard- 
miles- 


10.37 


feet. 


1.422 


acres. 


0.874 
0.65 


yard. 


2.598 


(( 


0.874 


i( 


0.865 


(( 


1.711 yards 
1.754 ** 


1.093 yards 
0.621 mile. 


0.749 
0.761 


yard 


2. 


t( 



FOREIGN LINEAR AND SURFACE MEASURES. 



a 15 



Foreign. 
Mechlin : 1 aune, - - - -^ 

BERMUDAS I. — Same as Great Britain. 
BOURBON I. —-3 pied= 1 aune, 
BRAZIL. — 12 poUegada = 1 pe, 5 pes = 1 passo, 
52 passi = 1 estadio, 24 estadi = 1 milha, 3 
milhe = 1 legoa, - - • - = 

8 pollegada = 1 palmo, 5 palmi = 1 vara, 2 
vare, or 3^ covadi, or 1^ passi = 1 bra9a, = 

1 geira, - - - - - = 
Bahia, Rio Janeiro ; 3 palmi = 1 covado, - = 

Central and South America. 

Balize, Bolivia, Buenos Ayres, Chili, Equador, 
GuATiMALA, New Granada, Peru, Uruguay, 
Venezuela, Yucatan : 
Nomenclatures and legal values, same as Castile 

( Spain) . 
Guiana. — Berbice, Demerara, Essequiho, Surinam: 

Same as Holland. 
Cayenne. — Same as France. 
CANARY I. — 12 onza= 1 pie, 3 p. = 1 vara, = 

2 vara = 1 braza, - - - - = 
52 braza cuadrada=!l celemin, 12 c. = l fa- 

negada, - - - - - = 

CANDIA I. — 8 robi = 1 pic, - 
CAPE COLONY. — Same as Great Britain. 
CAPE YERDE I. — Same as Lisbon (Portugal). 
CHINA. — 10 fan=l tsun or punt, 10 tsun=l 
kong-pu or chik, 10 kong-pu = 1 cheung, 
10 cheung = 1 yan, 18 yan = 1 li, - - = 

Chik (mathematical) = 1.094 ft. Chik (en- 
gineers''), - - - - = 
Chik (tradesm£n^s) = 1.21S ft. Kong-pu, - = 
l^- chik (engineers^) = 1 thuoc, 3^ thuoc 

=i po, = 

10 punts = 1 covid or cobre, If c. = 1 thuoc 
(mercers'*), - - - - = 

Pekin : 10 chik (math.) = 1 cheung, - - = 

CYPRUS I. — 8 robi = 1 pic, 
DENMARK. — 24 tomme or 2 fod = 1 aln, - 

3 aln = 1 favn, 1| f. = 1 rode, 2400 r. = 1 miil, = 
96 album 'or 8 sMepper = 1 toende, - - = 

EGYPT. — 2 derah = 1 fedan, 3 f. = 1 
8 rob = 1 pic, - - - 



U. States. 
0.753 yard. 

1.298 *' 



3.836 miles. 

7.214 feet. 
1.428 acres. 
0.713 yard. 



0.920 yard. 
5.522 feet. 



0.5 
0.697 



acre, 
yard. 



0.346 mile. 

1.058 feet. 
1,014 " 



yard, 
feet. 



5.025 

0.711 
10.937 

0.696 yard. 

0.688 *' 

4.681 miles. 

5.45 acres. 

12.67 feet. 

0.74 yard. 



16 a FOREIGN 


LINEAR AND 8URPA0B 


MEASURES 






Foreign, 








U. States. 


1 fedan al risach, 


- J - 




. = 


4. 


acres. 


Alexandria^ Rosetta 


; 1 pic stambuli, 


- 


= 


0.733 


yard. 




Pic for muslins, &c., 




- = 


0.G86 


i( 




Pic for cloths, 


- 


= 


0.613 


n 


FRANCE. — 100 centimetres or 10 decimetres = 


:1 







metre, - - - - - = 

100 metres or 10 decambtres = 1 hectometre, = 
100 hectometres or 10 kilometres = 1 myria- 
metre, - - - - - = 

100 square metres = 1 are, 100 a. = 1 hectare, = 
3j(j pied metrique= 1 aune = 47^ inches, - = 
GERMANY.— Baden (legal): 20 zoU or2fu8 = 
1 elle, - - - - = 

5 elle = 1 rutho = 3 metres of France, - = 
2 stunden = 1 nieile, - - - - = 

1 jauchart= 0.82 acre. 1 morgen, - = 

Manhcim : 1 fus = 0.952 ft. 1 elle, 
Bavaria (legal) : 120 zoU or 10 fus=l ruthe, = 
2400 ruthe = 1 meile, - . . = 

34j-zoll = lelle, 
1 jauchart or mor^^en, - - - = 

5 cubic fus= 1 khifter= 110.62 cubic feet. 
Augsburg : 2 fus = 1 elle, - - - = 

1 elle (mercers'') , - - - - = 
Nuremberg: 2^ fus = 1 elle, - - - = 
IIanover (bgal) : 12 zoll = l fus, - = 

2 fus = 1 elle = 0.038 yard. 8 e. = 1 ruthe, = 
14G22- ruthe = 1 moile, - - - = 
2 vierling= 1 morgen, - - - = 

Bremen : 24 zoU or 2 fus = 1 elle, - - = 

6 fus = 1 klafter, 2j k. = 1 ruthe, - = 
20000 Rhinehind fus = 1 mcilc, 

120 square ruthe = 1 morgen, - - = 

1 reif = 96.52 cub. ft. 1 faden = 61.6 cub. ft. 
Emden^ Osnaburg : 2| fus= 1 elle, - - = 

Hesse Cassel. — 24 zoU or 2 fus = 1 elle, - = 

14 fus = 1 ruthe, - - - - = 

1 klafter = 126.089 cubic feet. 
Hesse Darmstadt (legal) : 100 zoll or 10 fus = 
1 klafter = 2i /fillres of France, - = 

32 zoll = 1 elle," 

400 square klafter or 4 viertel = 1 morgen, = 
Frankfort : 12 zoll = 1 fus or werkschuh, - = 

2fus=lelle,2e. = l stab, - 

10 feldfus= 1 ruthe, - - - = 



1.094 

19.883 



rods. 



0.214 miles. 
2.471 acres. 
1.312 yards. 



0.656 
3.281 
5.524 
0.889 
0.610 
9.575 
4.352 
0.911 
0.841 

0.648 
0.666 
0.718 
0.943 

15.328 
4.246 
0.647 
0.633 

15.188 
3.S96 
0.636 



miles, 
acre, 
yard, 
feet, 
miles, 
yard, 
aero. 

yard. 



foot. 

feet, 
miles. 

acre, 
yard. 

feet, 
miles. 

aero. 



0.698 yard. 
0.«23 '* 
13.088 feet. 



8.202 feet. 
0.875 yard. 
0.618 acre. 
0.934 foot. 
1.245 yards. 
11.672 feet. 



FOREIGN LINEAR AND SURFACE MEASURES. 



a 17 



Foreign. 
HoLSTEiN. — Hamburg, Altona : 

24 zoll or G palm or 2 fus = 1 elle, 
3 cllc= 1 klafter, 2^ k. = 1 marschruthe, 
2| klafter (10 fus) = 1 geestruthe, 
24000 RhinelaDd fus (2000 R. ruthe) = 1 
mcile, - - - - _ 

1 Brabant elle for woollens, 
600 square marschruthe = 1 morgen, 

Lubcc : Denominations and relative values, same 
as at Hamburg. — 1 elle, 

Mecklenburg. — Rostock, <^c. — Same as Ham- 
burg. 

Saxony. — Dresden ^ Lcipsic: 12 linie = 1 zoll, 12 
zoll = 1 fus, 2 fus = 1 elle, 

2 elle = 1 stal), 4 stab = 1 ruthe, 

3 elle = 1 klafter, 

1500 ruthe = 1 meile, _ - . 

• 300 S(][uare ruthe = 1 acker, 
Freyburg : 2 fus = 1 elle, 5 e. = 1 ruthe, 
Oldenburg. — 24 zoll or 2 fus^= 1 elle, 

9 elle = 1 ruthe, 1850 r. = 1 meile, - 
GREAT J^RITAIN. — Same as United States. 
GREECE. — Patras : 8 robi = 1 pic/or silks, 

1 pic /or looollcns, c|-c., 
HOLLAND {legal): 10 streep = l duim, 10 d. = 
1 palm, 10 p. = 1 el = 1 metre of France, 

10 el = 1 roed, 100 r. = 1 mijl. 
Previous to 1820 — 2 J fus = 1 el, - 

El of Flanders, - - 

Hague — Brabant el , 



U. States. 

0.6266 yard. 

13.159 feet. 

5.013 yards. 

4.68 miles. 

0.761 yard. 

2.385 acres. 

- = 0.63 yard. 



0.618 yard. 
4.943 '' 
5.561 feet. 
4.213 miles. 
1.515 acres. 
9.619 feet. 
0.648 yard. 
6.133 miles. 

0.694 yard. 
0.75 

1.093 '' 
0.621 mile. 

0.747 yard. 
0.776 '' 
0.761 *' 



India and Malaysia or East Indies, 



An-nam. — Same as China. 

BiRMAH. — 4 taiin = 1 sadang, 7 s. = 1 bambou, = 4.208 yards. 

Ceylon I. — Colombo: 5 palmi = 1 covid, - = 0.516 " 
HiNDOSTAN. — Bombay : 2 tussoo = 1 gheria, 8 g., 

= 1 haut or covid, - - - -= 0.503 " 

Uhaut:^lguz, - - - = 0.755 *« 

Calcutta : 3 jaob = 1 anguUa, 3 a. = 1 gheria, 

8 g. = 1 haut or covid, 2 h. = 1 ghcs or guz, = 1. ** 

3 palgat= 1 hand, 5 h. = 1 cubit, - -= 1.25 foot. 

3 cubits = 1 corah, 1728 c. = 1 coss, - = 1.227 miles. 

Gfoa; 12pollegada=lpe, - - -= 1.082 feet. 



18 a 



FOREiaN LIN£AB AND SURFACE MEASURES. 



Foreign. 
24f pollegada = 1 covado avantejado, 
13^ pollegada = 1 terca, 3 t. = 1 vara, 
Madras : 8 gheria = 1 covid, 

1 casseney or cawney, - . . 

Massulipatam, 2 palm = 1 span, 3 s. == 1 cubit, 
Mysore, Seringapatam : 8 gerah = 1 haut, 2 h. 

= lgugah, .... 
Fondicherry : 8 gheria = 1 haut or covid, 
Sural: 84 tussoo or 20 wiswusa = 1 wusa, 

18 tussoo = 1 cubit or haut/or matting^ '- 
Tatta : IG garca= 1 guz, - - . 

Tranquehar, Scrampore {legal) : same as Den- 
mark, 
Java I. — Bnfavia : 8 gheria = 1 covid, cubit or el, 

1 fus {R/unish), - - . . 
Malacca. —' Malacca : 8 gheria = 1 covid, - 

8 covid = l jumba, - . ' . 

PiiiLipriNE I. — Luzon L — Manilla. — Same as 

Cadiz, Spai7i. 
SiAM. — 12 iiion = 1 keub, 2 k. = 1 sok, - 

2 8ok= 1 ken, 2 k. = 1 vouah, 20 v. = 1 sen 
40 8. = 1 jod, 25 jod = 1 roencng, 

Bangkok : 8 gheria = 1 cuvid, - 
Singapore I. — 8gheria=l hasU, 
Sumatra I. — 4 tempoh or 2 jankal= 1 etto, 2 etto 
= 1 hailoh, - - , . 

IONIAN ISLANDS. — 07>Wonia, Corfu, Ithaca, 
Pojros, Sf. Maura, Zantc, &c. : 
12 onue = 1 pie, 5 pe8= 1 passo, - = 

1 braccio ybr silks, - - . . __ 

1 braccio/<:>r woollens, - - - _ 

1 moggio (/i/ifffr), - - . . __ 

30 inchois or 3 feet=l yard, - - = 

ITALY. — Lombardv and Venice : 
Government and Customs Measure — 
10 atome= 1 dito, 10 d. = 1 j^uilmo, 10 p. = 1 

metro or braccio = 1 metre of France, - = 

1000 metre =1 miglio, - - - = 

100 square metre = 1 tavola, 100 t. =1 torna- 
tura, - - - . . __ 

Special a fid local — 

Ve?iic€ : 2 palmi = 1 braccio, 2i b. = 1 passo, = 
li passi = l portica. 

4.i pede — 1 chebbo, Ij c. = 1 cavezzo, - = 

1 passo, geometrical, - . . :^ 

1 braccio ybr woolle/is, - - . = 



u. 


Slates. 


0.744 


yard. 


1.203 


(( 


0.615 


(( 


1.32 


acres. 


1.594 


feet. 


1.072 


yards. 


0.5 


a 


2.712 


(( 


0.581 


(t 


0.943 


<< 


0.75 


(( 


1.03 


feet. 


0.5 


yard. 


12.— 


feet. 



= 0.525 yard. 

= 2.388 miles. 

= 1.5 feet. 

= 1.5 

-= 1.03 yard. 



5.455 feet. 
0.705 yard. 
0.755 *« 
2.4 miles. 
1. yard. 



3.281 feet. 

0.621 mile. 

2.471 acres. 

5.699 feet. 

6.845 ** 

4.559 ** 

0.739 »' 



FOREIGN LINEAR AND SURFACE MEASU&8B. <Z 19 

Foreign. U. States. 

1 braccio, ybr silks, - - - =3= 0.693 feet. 

Naples. — 5 minuto= 1 oncia, 12 0. = 1 palmo, 8 

palmi= 1 canna, - - - - = 6.92 " 

7i palmi = l passo or pertica, 8 pertica=l 

catena, 11 6| catene=a= 1 miglio, - = 1,147 milea. 

900 square passi = l moggio, - - -= 0.87 acre. 

1 braccio (2| palmi in theory), - « = 0.764 yard. 

Sardinia. — Genoa: 8 oncie = l pie, 10 p. or 12 

palmi = 1 canna (5wrt'eyor5'), - -= 9.715 feet. 

2j palmi = 1 braccio, - - - = 0.63 yard. 

9 palmi = 1 canna picolo, - - -5= 2,429 " 

10 palmi = 1 canna, ybr linens, 
12 oncia = 1 pie liprando. 

Nice : 12 oncia = 1 palmo. 25 oncia = 1 raso, = 0.600 " 
Turin: 8 oncia = 1 pie manual, - - =1.19 feet. 

12 oncia = 1 pie liprando. 

14 oncia =1 raso, - - - -= 0.649 yard. 

5 pio manual = 1 tesa. 

6 pie lip. = 1 trabucco, 2 t. = 1 pertica. 

States of the Chruch. — Ancona : 1 braocio, = 0.704 *' 
10 pie =- 1 pertica, - - - =13.438 feet. 

Rome: 10 decime or 5 minuto= 1 oncia, 16 0. 

= 1 pie, 5 piede = 1 passo, - - = 4.884 feet. 

5 linea = 1 parto, 24 p. = 1 palmo, 8 palmi = 

1 canna, - - - - = 2.176 yards. 

Tuscany. — Leghorn, Florence, Pisa: 

12 denari or 3 quattrini = 1 soldo, 20 soldi or 2 

palmi = 1 braccio, 4 b.=:l canna, - -= 2.552 *' 

8 braccia=:l passo, 2 p. = 1 cavezzo, - zz 11.484 feet. 

5 braccia=: 1 pertica, 5GG| p, === 1 miglio, = 1.027 miles. 

JAPAN. — 5 Kupera sasi = 1 ink, - - - = 2.072 yds. 

l\ sasi = 1 k. sasi, 2^ sasi = 1 ikje, - - = 2.32 feet. 

MALTA I. — 12 oncie = 1 palmo, 8 palmi or 7j 

piede = 1 canna, - - - - =: 2.275 " 

MADEIRA I. — Standard same as Lisbon, Port- 

ugaL 
MAURITIUS I. — Standard same as Great Brit- 
ain, 
MEXICO, — Same as Cadiz, Spain, 

MOROCCO. — iV%f^r/ore: Icadee, - - = 1.695 feet. 

1 covado = 1.654 feet, 1 pic, - -= 0,723 yard. 

NORWAY, — Same as Denmark. 

PERSIA. — 1 archin arisch, - - - = 1.063 " 
1 arcbin schab, - - - - =b 0.874 " 

1 gueza {royal) = 3xV fi^^* 1 gueza {com- 
man), « 0.092 *' 



20 a 



FOREIGN LINEAR AND SURFACE MEASURK. 



Foreign. 

1 monkelzer, - - - - - = 
Bushire : 1 guz, - - - - = 

PORTUGAL. — Lw^on, St. Vbes, cfc. ; 

80 pollogada, or 10 palmi do craveira, or 6| 
pes, or 3 J covado, or 2 vara, or IJ pa880 = l 
Dra^a, - - - - - = 

780 p8=s 1 estadio, 8 e8tadi = l milha, - = 
3 millia = 1 legua. 

8 pollogada = 1 palnia, 3 p. = 1 covado, - = 

24| polU'gada = 1 covado avantojado, - = 

13J pollogada = 1 terca, 3 t. =1 vara, - = 

Oporto: 3j)iilina = l covado, - - = 

PRUSSIA (Irynl since 1S20) : 

12zolle=l fuH (Rhcin-fu8), - 

10 zolle= 1 land-fuH, 10 land-fus or 12 Rhine- 

fii8= 1 null, 2(MM) r. = 1 incilc, 
254 zollo (Uhcin-zollc) = l flic, 
180 Htjuan* ruthe= 1 iiiorgon. 
Dantzic (sp(ci(t^) : 75 am = 1 i!»cil, - - = 

Konig$bcr(j : 1 die, - - - - = 

RUSSIA (Icyalfor the Empirv) : 

IG vcr8chok= 1 archine, - - - = 

3 archinos or 7 feet = 1 sachino, - - = 

500 8Jicliinc8= 1 vcrst, - - - = 

24(K) Hfjuaro 8acln!U'= 1 dcciatine, - - = 

Crimka. — Strastopol, cfr. ; 1 halcbi, - - = 

SARDINIA I. — 12 oncia=i 1 palina. - 

22 oncia = 1 pic, - - - - = 

25i oncia = 1 riu*o, - - - = 

12 jmlini=l trahucco. 10 palmi = 1 canna, = 

SICILY I. — Mtssina: 8 palmi = 1 canna, - = 

Pahrmo : 8 j)alnii = 1 canna, - - - = 

SPAIN. — Alkant : pulga<la := 1 palnio, 4 palmi 

= 1 vara, 2 vara = 1 hraza, - - = 

12 pulgada or l^ jialmi = 1 pic, - - = 

Darcrlona : 4 j)alini = 1 vara or matja-cana, = 

2 vara = 1 cana, - - - - = 
Cadiz {Standanl of Castile) : — pulgtida = 1 

Bcsma, 2 8. = 1 pie or tcrcia, 3 pic = 1 vam, = 
5 pio = 1 pi8So. 

2 octava = 1 quarta or palma, 4 q. = 1 vara, = 
12 pulgada = 1 pie, IJ i)ic = 1 eodo, 2 c. = 1 
vara, 2 v. = 1 estado, braza, brazada or 
tocsa, 2 c. = 1 cucrda, 2 c. = 1 cordel, 500 
cordclo = 1 Icgua, - - - - « 



U. States. 
2.351 feet. 
0.557 yard. 



7.214 


ftH3t. 1 


1.27DmiKij. 1 


0.722 


yard. 


0.744 


(i 


1.203 


ik 


0.707 


a 


1.03 


fwt. 


4.08 


miles. 


0.7203 


yard. 


47.002 


»( 


0.G28 


it 


0.777 


ti 


7. 


foot. 


o!gg3 


mile. 


2.7 


acn>8. 


0.709 


yard. 


0.286 


it 


1.571 


feet. 


O.G 


yard. 


2.87 


it 


2.311 


ti 


2.07 


(i 


1.G77 


it 


0.833 


foot. 


0.849 


yard. 


5.094 


foct 


2.782 


(i 


0.928 


yard 


4.215 milo6. 

1 



FORig[aN LINEAR AND SURFACE MEASURES. a 21 

Foreign, U. States, 

192 vara cuadrada = 1 quartillo, 4 q. = 1 ce- 
lemin, 7^ c. = 1 aran^ada, 1| a. = 1 fane- 
gada, - - - - - = 1.587 acres. 

50 fanegada = 1 yugada. 
Corunna, Ferrol : 4 palma or 8 octava= 1 vara, = 0.928 yard. 
Gibraltar. — As at Cadiz ; also, as in Great Britain, 
Malaga. — Same as Cadiz, 

Santander : 8 octava or 4 palma = 1 vara, - = 0.913 " 

Valencia : 9 onze= 1 palmo, li p. = 1 pie, = 0.992 feet 

3 pie = 1 vara, 2^ v. = 1 braza reale, - = 2.232 yards, 
SWEDEN. — 12 tum = 1 fot, 2 f . = 1 aln, - = 0.G48 ** 

3 aln= 1 famn, 2§ famn = 1 stang, - = 15.553 feet, 
2250 stang= 1 mil, - - - - = G.627 miles, 

4 kappland = 1 fjerding, 4 f. = 1 spannland, 2 

s. = 1 tunnland = 218} square stang, - = 1.211 acres, 
SWITZERLAND. — Legal, since 1823, for the 
Cantons of AaraUj Basle, Berne, Freiburg, 
Lucerne, Solothurn, Vaud ; but not in gen- 
eral use : 
10 zoU = 1 fuss, 4 f . = 1 stab = 1 aune of 

France, - - - - -= 1,3124 yards, 

24 stab = 1 toisc or ruthe, - - = 3.2809 *' 

Special and local — 

Basle : 12 zoll = 1 schuh or fuss, 10 e. «= 1 ruthe, = 3.33 '« 

21i zoll = 1 braccio, . - - =» 0.5966 ** 

444 '* =lclle, =1.2348 ** 

464 '' =lklafter. 
Berne : 12 zoll = 1 fuss, 6 f. »= 1 klafter, 1| k, <?r 

10 fuss=l ruthe, - - - -=9.6215 feet, 

m fuss = l elle, - - - - = 0.5933 yard. 

Geneva : 12 zoll = 1 pied, 5 J p.= 1 toise, - = 8.528 feet. 

1 aune (wholesale), - - - -= 1.299 yards. 

1 aune (retei7) , - - - - = 1.25 *« 

Lausanne : 3f piede = 1 aune (metrical, of 

France), - - - - - = 1.3123 " 

9 piede = 1 ruthe. 
Neufchatcl : 12 zoll = 1 fuss, 10 f . = 1 tois, - = 9.621 feet. 
22i zoll = 1 elle, - - - - = 0.608 yard, 

St. Gall: 10 zoll = 1 fus, 4 f . = 1 stab, - = 1.3123 *' 
Zurich : 12 zoll = 1 fus, 2 f . = 1 elle, - - = 0.6561 " 

5 elle = 1 ruthe. 

TRIPOLI (N. Africa) : 8 robi = 1 pic, - = 0.6041 " 

1 pic /or ribbons, - - - - = 0.5285 " 

TUNIS. — 8 robi = 1 pic, for woollens, - = 0.736 " 

1 pic, for silks, - - - - = 0.690 " 

1 pic, for linens, - - - - = 0.5173 " 



22 a FOREIGN LINEAR AND SURFACE MEASUI^pS. 



Foreign, 




U. States. 


TUKKEY. — AUf^o : 8 robi = 1 


pic. 


- = 0.7396 yard. 


1 dra meifl'our, 




= 0.G089 ** 


1 dra stambouli, 


- 


- = 0.7079 *' 


Bagdad : 1 guz, 


- 


= 0.8796 »* 


Bussorah: 1 guz = 2.6389 ft. 


1 hadid, 


- = U.9502 *' 


Constantinople: 1 halebi, 


- 


= 2.325 feet. 


1 endrasi or archim, - 


- 


- = 2.2'')5 *' 


1 pic stambouli, - 


- 


= 0.7079 yard. 


1 pic for silks. 


- 


-=0.7317 ** 


Damascus : 1 pic, - 
Smyrna : 1 indise, 


- 


= 0.(k^7 " 


- 


- = 0.6846 " 


8 rob = 1 pic, 


- 


= 0.7302 '* 


West 


Indies. 





In the islands of Antigua, the Bahamas, BarbadoeSy 

Barbuda, Dominicaj Grenada, Jamaica, Les 

Saints, Montserrat, Nevis, Santa Cruz, St. 

John, St. Kitts\ St. Thomas, St. Vincent y 

To^)ago, Tor tola, the Measures of Length are 

the same as in Groat Britain. 
In Deseade, Guadeloupe, Mariegalantc, Martiniquey 

St. Lucia: 
12 ponce = 1 pied de Roy, 3^ p. = 1 aunc, - = 1.3 yards. 
This being the old system, of France', or system 

previous to 1812. 
In Bonaire, Saba, St. Eustatius, Si. Martin, Somo 

as Holland. 
In St. Bartholomeio, Same as Sttedkn. 
In Curacoa, Trinidad, Same as Castile (Spain). 
Cuba I. — Cardenas, Cienfuegos, Havana, Matan- 

zas, Nuevitas, Porto Principe, St. Jago, &c. : 

Same as Castile (Spain). 
Hayti I. — Aux Caycs, Cape Haytien, Jcrcmic, Port 

au Prince, Port Platte, &c. : Same as France, 

before 1812. 
Savanna, St. Dmingo, &c. : Same as Castilb 

( Spain) . 
Porto Rico 1. — Ponce, St. Johns, &c. : Same as 

Castile (Spain). 



FOREIGN WEIGHTS REDUCED TO UNITED STATES. 



Foreign. U. States, 

Avoirdupois 
pounds. 
ABYSSINIA. — Masmah : 10 dirhcm = 1 wakea, 12 

w. or 10 mocha = 1 rotl or liter, - - = 0.688 

ALGIERS. — 8 initkal = 1 wakea, 16 wakea = 1 rotl 

SLttar'i (for spices and drur/s), - - = 1.190 

18 wakea = 1 rotl ghoddari {for fruits, oil, &c.), = 1.339 
27 wakea = 1 rotl khebir (market pound) , - = 2.008 

100 rotl = 1 cantaro or quantar. 

1000 grammes = 1 kilogramme, - - = 2.205 

ARABIA. — Ilodcida : 30 vakia = 1 maon, 10 maon = 

1 frazil, 40 frazils = 1 bahar, - - - = 813. 

Jidda: 15 vakia = 1 rotolo, 5 rotoli = 1 maund, = 1.83 

10 maunds = 1 frazil, 10 f. = 1 bahar, - = 183. 

Mocha: 15 vakia or wakega = 1 rotolo or rotl, - = 1.5 

2 rotolo = 1 maund or maon, 10 maunds = 1 frazil, 

15 frazils = 1 bahar, - - - - = 450. 

100 miscals = 1 vakia, 22^ v. = 1 maund copra, = 2.25 
At the Bazaar, for coffee — 

144 vakia = 1 rotolo, a7id a bahar, - - = 435. — - 

Muscat: 24 cucha = 1 maund, - - - =: 8.75 

AUSTRIA. — Trieste, Vienna: 2 lothe = 1 unze, 4 u. 
= 1 vierding, 2 v. = 1 mark, 2 m. = 1 pfund, 20 
p. = 1 stein, 5 s. = 1 centner, - _ = 123.47 

4 centner = 1 karch. 1 saum, - - - = 339.5 

1 saum^or steel, - - - - = 308.666 

Trieste (Venice weight) : \ \)^\in(}i (peso grosso) , -= 1.052 

1 pfund (peso sottile) — apothecaries^ weight, - = 0.665 
Bohemia. — 100 pfund = 1 centner, - - - = 113.4 

Prague: 16 unze = 1 pfund, - . - - = 1.26 

18 pfund = 1 stein, 6 s. = 1 centner, - - = 136.08 

Hungary. — 1 oka, - - - - - = 2.78 

AZORE ISLANDS. —Same as Lisbon (Portugal), 
BALEARIC ISLANDS. — Majorca. — 25 rotoli = 1 

arroba, - - - - - - = 22.37 

4 arroba = 1 quintal, 112 rotoli = 1 oder, - = 100.217 
100 rotoli barbaresco = 1 quintal, - - - = 92.794 



24 a FOREiaN weights reduced to united states. 

Foreign, U. Slates, 

Avoirdupois 
poauds. 

Minorca. — 100 libra = 1 cantaro, - - -=88.2 

100 rotoli barbaresco = 1 quintal, - - =: 81.727 

3 quintals = 1 carga, - - - - = 275.18 

BELGIUM. — Same as Holland. 
Previous to 1820 — 
8 pond = 1 stein, 12i s. = 1 centner, - - = 103.659 

3 centner = 1 schippond, 4 centner = 1 charge. 

Waeg of coals, - - - - - = 150. 

BERMUDAS I. — Same as Great Britain. 

BOURBON I. — 100 li\TC8 == 1 quintal, 3 q. = 1 charge, = 323.765 

BRAZIL. — IG onc;a = 1 arratel, 32 arratid = 1 arroba, = 32.501 

4 arrobas = 1 quintal, 13i q. = 1 tonelada. 

Central and South Ameiica, 

Balize, Campechc, Gxiatimala, Honduras, Laguna, Leon, 
Nicaragua, San Juan, San Salvador, Sisal, <^c. 

Buenos Ayns, Callao, Carthagena, Co(/ui/nI)o, Guaya- 
quil, Lagvayra, Lima, Maracayho, Montevideo, Rio 
Hacha, Tru.villo, Valparaiso, &c. : 
16 onza = 1 libra, 25 1. = 1 arroba, 4 a. = 1 quin- 
tal, = 101.546 

Berbice, Doner ar a, Essequibo, Surinam, — Same as 

Il0LL.\ND. 

Cayenne. — Same as France. 
CANARY ISLANDS. — Same as Castile {Spain), 

CANDIA I. — 44 oka = 1 cantxiro, - - = 116.568 

CAPE COLONY.-- CVc Towm. — 32 loot = 1 pond, = 1.03 
CAPE VERDK ISLANDS.— Same as Lisbon (Por/?/^a/). 
CHINA. — 10 lis = 1 tacl or Icung, 10 taels = 1 catty 

or kan, 100 cattii's = 1 pecul or tarn, - - = 133.333 
22| chu = 1 leung, 16 1. = 1 catty, 2 c. = 1 yin, 

15 y. = 1 kwan, 3J k. = 1 tam, 1 J t. = 1 shik, = ICO. 

CORSICA I. — llivre, = 0.76 

CYPRUS I. — 400 drachmi = 1 oka, 40 oka = 1 moosa, = 112. 

15 oka = 1 rotolo, - - - - = 5.25 

DENMARK. — 32 lod = 16 unze = 2 mark = 1 pund, = 1.101 

100 pound = 1 centner, * - - - - = 110.11 

12 pund = 1 bismerpund, IJ b. = 1 lispund, 20 1. 

= 1 shippund, - - - - - = 352.364 

2.1 lispund = 1 waag, - - - = 39.639 
EGYPT. — 144 drachmi or 100 miscall = 1 rotolo for- 

fora, - - - - - -= 0.95 

400 drachmi = 1 harsela, - - - =z= 2.039 

420 drachmi = 1 oka, - - - - = 2.771 



S^OKEIGN WEIGHTS REDUCED TO UNITED 8TATE3. CL^ 

Foreign, U. States^ 

Avoirdupoig 
pounds. 

100 rotoli or 36 harseli = 1 cantaro ferfora, - = 95. 

105 *' or 36 oka = 1 cantaro,/(?r co^ee, - = 99.75 
1 votolo minn, for spices, - - - -= 1.403 

1 '* zsiidmQ, for dye-woods J - - = 1.138 

1 <* zauro, for iron^ - - - - = 2.216 

102 rotoli == 1 cantaro j/^r quicksilver and vermilion, = 96.9 
• 115 *' =1 << for almonds and fruit, - = 109.25 
125 '< =1 ^' for drugs, - - -=118.75 

133 *' =1 '' forgumaraUc, - =126.35 

150 <' =1 ^< for plumbago, - -=142.5 

FRANCE, — 1000 milligrammes = 100 centigrammes = 
10 decigrammes = 1 gramme = 15.43315 troj 
grains, 
100 grammes = 10 decagrammes = 1 hectogramme, = 0.22 
10 hectogrammes = 1 kilogramme, - - = 2.205 

10 kilogrammes = 1 myriagramme, - - = 22.047 

100 myriagrammes = 10 quintal = 1 tonneau. 
1 livre poids de marc, - - -' -=1.07922 

1 livre mctrique = 4 kilogramme, - - = 1.10237 

GERMANY.— BAJ)Ey\— Heidelberg, Manheim, &G. : 
1000 as = 100 pfennig = 10 centas = 1 zehnling. 
1000 z. = 100 pfnnd = 10 stein = 1 centner, - = 110.237 
8 quentchen = 2 loth = 1 unze, 

16 unze = 2 mark = 1 pfund, - - = 1.102 

Bayaria. — 20 pfund (legal) = 1 stein, - - = 24.693 

Augsburg : 512 pfennig = 128 quentchen = 32 loth 

= 16 unze = 2 mark = 1 pfund or frohngewicht, = 1.082 
100 pfund = 1 centner, - - - - = 108.262 

3 centner = 1 pfundschwer or schiffpfund, - = 324.786 
74 pfundschwer <?r 100 stein = 1 tonne. 

224 pfund = 1 stein, 5 J stein =: 1 wage, - - = 129.914 

Nuremberg : Denominations and relative value, same 
as Augsburg, 
100 pfund = 1 centner, - - - = 112.432 

Hanover (legal) : 32 ortchen = 16 drachma or quent- 
chen = 2 loth = 1 unze, 8 u. = 1 mark, 2 m. = 
1 pfund, - - - - - = 1.079 

14 pfund = 1 liespfund, - - - =15.11 

24 liespfund = 1 pfundschwer or last, - - = 362.648 

20 pfund = 1 stein /ar/«.r, - - - = 21.586 

Brefmen : 32 ortchen = 8 quentchen == 2 loth = 1 unze, 

8 u. = 1 mark, 2 m. = 1 pfund, - - = 1.099 

116 pfund = 1 centner, - - - - = 127.516 

24 centner = 1 schiffpfund, - - - s=: 318.791 

c 



26 a FOBEIGN WEIGHTS REDUCED TO UNITED STATES. 



II 



Foreign, U. States. 

Avoircluiwia 
poviiKls. 

300 pfand = 1 pfundschwer or last, - - — 321). 784 

14 pfund =^ 1 liespfund. 1 wage, /or iron, - = 131.913 

1 BiQin Jar flax = 21.985 lbs. 1 stein, /or wool, = 10.993 

Emden, Osnaburg : 16 unze = 2 mark = 1 pfund, 100 

pfund = 1 centner, - - - - = 109.04! 

3 centner = 1 pfundschwer. 
Hesse Cassel. — 10 unze = 1 pfund, 100 pfund = 1 

centner, - - - - - = lOG.TGl 

21 pfund = 1 kleuder. 
Hesse Darmstadt {legal) : 100 pfund = 1 centner, = 110.23( 

Frankfort: IG unze or 2 mark = 1 pfund, - - = 1.11 

100 pfund = 1 centner, - - - =111 .4()| 

22 pfund = 1 stein, - - - - = 24.501 
HoLSTEix. — Hamburg, Liihcc, Altona, Kiel: 

IG unze = 2 murk = 1 pfund, - - - = l.OG^ 

14 pfund = 1 Ih'spfund, S 1. = 1 centner, - = 119. G 
24 centner = 1 schiffpfund, - - - = 299.-— 

7f schiffpfund or 100 stein = 1 tonne, - = 2135.7: 

90 pfund = 1 hist, - - - - = 9G.11 

1 stein, for feathers and icool, - - - = 10.G7J 

320 pfunds {landfrnght) = 1 schiffpfund, - = 341.72 

Mecklenburg {generally) : IG unze = 2 mark = 1 
pfund, 15 pfund = 1 liespfund, 20 1. = 1 schiff- 
pfund, ...-.= 313.721 
Bostock : IG pfund = 1 liespfund, 20 1. = 1 schiffpfund, = 358 54 
280 pfund == 1 schiffjifund,/or iron and lead, - = 313.721 
22 pfund = 1 stQin, J or xcool and flax, - - = 24.05 
Saxony. — Dresden, Lripsic: 32 pfennig ==8 drachma or 
quentlein = 2 loth = 1 unze, 8 u. = 1 mark, 2 
mark = 1 pfund, - - - -=1.03 
22 pfund = 1 stein, 2 s. = 1 wage, - - = 45.324 
110 pfund = 1 centner, - - - - = 113.31 
114 '* = 1 '< berg-gewicht. il 
118 *' =1 ** stahl-gewicht. -• 
Freyherg : 1 pfund, - - - - =1.105 
Oldenburg. — Denominations and relative values, same 11 
as Bremen, but weight values 2.94% less. || 
GREAT BRITAIN.* — See United States. ^ ' 

* In Great Britain, in addition to the denominations of weights used in tlie United 
States (the values of which are the same), the 



Clove of wool, = 

Stone " *' iron, flour, . s= 

Tod « « ers 

Weigh " " = 



Last 



7 lbs. 


14 




28 




182 




364 




4363 





Stone of butchers' meat or flesh, = 8 lbs. 

Stone of cheese, = 16 ** 

Stone of glas?, = 5 ** 

Seam of glass, = 120 " 

Stone of liemp, = 32 ** 

Fothcr of lead, .... • . . . = 19i cwU 



FOREIGN WEIGHTS REDUCED TO UNITED STATES. a 27 

Foreign. U, States, 

Avoirdupois 
pounds. 

GREECE. ■-'Athens : 400 drachmi = 1 oka, - - = 3.37 

44 oka = 1 cantaro, - - - - = 148.3 

MoREA. — 1 ca-nt'dTO, generally, - - - -= 123.75 

Fatras : 400 drachmi = 1 oka, - - - = 2.643 

44 oka = 1 cantaro or quintal, - - - = 116.3 

HOLLAND (legal) : 10 korrel = 1 wigtje, 10 w. = 1 

lood, 10 1. = 1 onz, 10 0. = 1 pond = 1 kilogramme 

of France, - - - - -= 2.204 

2000 ponds = 1 vat (shipping), - - =2204.74 

Previous to 1820 — 

300 ponds = 1 schippond, - - - - = 326.77 

8 ponds = 1 steen, - - . - _ = 8.71 

India and Malaysia or East Indies, 

An-nam. — Cochin China — Saigon : 16 luong = 1 can, 

10 can = 1 yen, 5 y. = 1 binh, - - - = 68.876 

2.binh = 1 ta, 5 ta = 1 quan, - - = 688.76 

Tonquin — Kesho : 100 catties = 1 pecul, - - = 132. 

BiRMAH. — Pegu : 12^ tical = 1 abucco, 2 a. = 1 agito, 

4 agiti = 1 vis, - - - - = 3.393 

33 tical = 1 catty, 3 c. = 1 vis, - - - = 3.393 
Rangoon : 2 small rwes = 1 large rwe, 4 large rwes = 
1 bai, 2 b. = 1 mu, 2 m. = 1 mat'h, 4 mat'ha 

s= 1 kyat or ticul, 100 k. = 1 paitktha or vis, = 3.65 

Borneo I. — 100 catty = 1 pecul, - - - = 135.633 

Ceylon I. — Colombo : 500 pond = 1 bahar or candy, = 500. 

Celebes I. — Macassar: 100 catty = 1 pecul, - = 135.633 
HiNDOSTAN — Bengal, generally (bazar weight) : 

10 mace*= 1 klianclia, 3 k. = 1 chattac, 10 c. = 1 

dhurra or pussaree, 8 d. = 1 maon or maund, = 82.133 
Calcutta (factory iveight) : 5 sicca = 1 chattac, 16 c. 

= 1 seer, 40 s. = 1 maund, - - - = 74-666 
Bombay : 36 tanks or 15 pice = 1 tippree, 2 t. = 1 

seer, 40 s. = 1 maund, 20 m. = 1 candy, - = 560. 

2^ tank-seers = 1 rupee-seer, for liquids = 1.54 lbs. 

Goa : 32 seers = 1 maund, 20 m. = 1 bahar, - = 495. 

Madras: 10 pagodas or varahuns = 1 pollam, 8 p. = 
1 seer, 5 s. = 1 vis or visay, 8 v. == 1 maund or 

maon, - - - - - - = 25. 

20 maunds = 1 candy or baruay. - - = 600. 

Malabar Coast : 40 polams = 1 vis, 8 v. = 1 maon, = 30. 

20 m. = 1 candy, 20 c. = 1 garce. 

Malabar (interior) : 20 maon = 1 candy, .- - = 695.— 



28 a FORHGIf WEIGHTS REDUCSB TO UIOTED STATBl 

Foreign, U. States, 

Avoinlupoi9 
pouDdB. 

Mangalore : 6 Bida= 1 vis, 8 v. = 1 maund, 20 maunds 

= 1 candy, ---.-= 504.72 
MassuUpatam : IJ nawtauk = 1 chittac, G c. = 1 
yabbolam, 2 y. = 1 puddalum, 2i p. =• 1 vis, 6| 

V. = maund, 20 m. = 1 candy, - - - = 500. 

21 J vis = 1 pucca maund, - - - = 80. 

Ml/so re, Scringapatmn : 10 varahuns =• 1 pollam , 40 

p. = 1 puHsaree, 8 pussarcc = 1 iiiaon (yr maund, = 24.27G 
20 maund = 1 bahar or candy. 
Pondichcrry : 10 varahuiLS = 1 poloin, 40 p. = 1 vis, 

8 v. = 1 maund, 20 m. = I candy, - - = 588 

Sframpore^ Trawjucbar (legal) : Siuiio lUJ Denmark. 
Sinde : 2 pice = 1 anna, 2 a. = 1 cliittac, IG c. = 1 

seer, 40 seers = 1 maund, - - ' - = 74.GGG 

Surat (new ineasxin) : 8 pico = 1 tipprcc, 2 t. = 1 

seer, 40 s. = 1 mauna, 21 m. = 1 ciuidy, - =» 300. 

3 candi = 1 bhaur. 
Tatta: 4 pice = 1 anna, 16 a. = 1 seer, 40 s. = 1 

miumd, - - - - - - = 74.32 

Java I. — Batavia : IG tacl = 1 catty, I4 c. = 1 j^oclak, 
GG5 g. or 100 catty = 1 pccuf, 3 p. = 1 Ixihar = 
IG m. 1 vis, 24 iK)llams of Madras, - - = 405.333 

44 pcoul = 1 p;reat bahar. 

5 pccul = 1 tiuib;in^,/t>r yraiTi, - - =075.555 

Bantam: 24 taol =» 1 gwlak, 100 g. => 1 catty, 2 

catty = 1 Iwihar, /or p^;);;rr, - - - = 405.33S 

Malacca. — Malacca : IG ta<4 = 1 catty, 100 c. = 1 po- 

cul, 3 pocuLs= 1 Ixiliaf, - - . =405.— 

24 pinga =■ 1 tampang, 2 t. = 1 Ixidoor, 12 b. = 1 

hali, l.i h. = 1 kip,/or /in, - - - = 40.G77 

2 buncals = 1 catty ,/or gold and silnr, - = 2.04'J 

PniLirriNE I. — Manilla, &c. : 

22 piiistrcs = 1 catty, 100 c. = 1 pecul, - = i;VJ.443 

1 caban of rice ( ?/5Ma/) , - - - -= 133. 

1 cal)an of cocoa, ---_=- 83.5 

Siam. — Bangkok, &c. : 4 tical = 1 tiiel, 2 t. = 1 catty, 

100 catty = 1 pecul, - - - - = 135.238 

Singapore I. — Same as Malacca. 
Soou)o IsiANDS. — 10 mace = 1 tacl, IG t. = 1 catty, 

50 c. = 1 lachsa, 2 1. = 1 pecul, - - :--i:::^:^33 

Sumatra I. — G2 catties = 1 pecul, - - -= lo-..387 

24 tael = 1 salup, 2 s. = 1 ootan, 7i o. = 1 nclli, 
for camphor, - - - - - ^ 29.333 



FOEEIGN WEIGHTS REDUCED TO UNITED STATES. • a 29 
Foreign, U. States, 

Avoirdupois 
pounds. 

Acheen : 10 mace = 1 tael, 20 t. = 1 goelak, 1| g. =: 

1 catty, 36 c. = 1 maund, - - - = 76.986 

5.J maund = 1 candil or bahar. 
Bencoolen: 46 catties = 1 maund, - - = 98.371 

5| maunds = 1 bahar, - - - - = 557.416 

IONIAN ISLANDS. — Cephalonia^ Corfu^ Ithaca, 
Paxos, Zante, Sec. : 

Legal since 1817 : 100 libbra = 1 talento, - = 100. 

100 oke = 385 marcs, 

44 oke = 1 cantaro, - - - - = 118.807 

Cephalonia : 64 libbra = 1 barile,/or salt, - = 67.262 

Corfu : 100 libbra = 1 talento, - - - = 90.034 

ITALY. — Carrara. — 1 carrata, for marble, = 25 

cubic palma = 12.764 cubic feet = 31f centi- 

najo or 29 J quintale of Modena, - - =2240. 

LOMBARDY AXD VenICE. 

Government and Customs Measure : 
10 grani = 1 dcnaro, 10 d. i= 1 grosso, 10 g. = 1 
oncia, 10 o. = 1 libbra, 10 1. = 1 rubbio, 10 r. 
= 1 centinajo = 10 myriagrammes of France, = 220.474 
Special and local : 

Venice. — Peso grosso : 4 grani = 1 earato, 32 c. = 1 
saggio, 6 s. = 1 oncia, 6 o. = 1 marco, 2 m. = 

1 lil)bra, = 1.052 

25 libbre = 1 miro, 40 m. = 1 migliajo, - - =1051.80 

Peso sottile : 4 grani = 1 earato, 24 c. = 1 saggio, 

6 s. = 1 oncia, 12 o. = 1 libbra, - - = 0.666 

100 libbre = 1 quintale, 4 q. = I carica, - - = 266.332 

Naples. — 20 acini = 1 trapcso, 30 t. = 1 oncia, 12 o. 

= 1 libbra, 20 1. = 1 rubbio, - - = 18.387 

150 libbra = 1 cantaro piccole, - - - = 106.08 

33 J onci = 1 rotolo, 10(3 r. = 1 cantaro grosso, = 196.45 . 
Sardinia. — Genoa : 24 grani = 1 denaro, 24 d. = 1 oncia, 
12 o. = l libbra, 1.^ 1. = 1 rotolo, 16| r. 
(25 libbre) = 1 rubbio, 4 r. = 1 centinajo, l.J c. = 

1 cantaro. 
Peso grosso : 1 centinajo, - - - =76.863 

Peso scarso : 1 centinajo, - - - - = 69.875 

Nice: 12 oncia = 1 libbra, 25 1. = 1 rubbio, 4 rubbi 

= 1 centinajo, _ - _ - = 68.694 

Turin: 25 libbre = 1 rubbio, - - - = 20.329 

States of the Church. — 1 libbra Italiana, - = 2.204 

Ancona: 12 oncia = 1 lira, 100 1. = 1 cantaro, - = 72.942 

c* 



80 a JOKELQJH WEIGHTS BEDUOED TO UNITED 8TAT19* 

Foreign, V, States. 

Avoidupoit 
pounds. 

Borne : 12 oncia = 1 libbra, 10 1. = 1 decina, 10 d. = 

1 cantaro, - - - - - = 74.703 
160 libbre = 1 cantaro. 250 libbre = 1 cantaro. 

1000 libbre = 1 migliajo, - - - «= 747.033 

TuscAxr. — Leghorn, Florence, Pisa : 

72 grani or 3 denari == 1 dramma, 96 dramme or 12 

oncia =« 1 libbra, 100 1. = 1 centinajo, - - = 74.857 

10 centinaje = 1 migliajo. 

160 libbre = 1 cantaro or carara, /or wool, fish , 4*C-> == 110.771 
50 rottoli = 1 cantiiro generale (old), - - =s 112.29 

JAPAN. — 160 rin = 10 pun = 1 its-go - - = 1.333 lbs. 
2,500 pun = 100 ischo = 10 itho = 1 its 'ko-koo = 333 J lbs. 
MALTA 1. — 30 traposi = 1 oncia, 30 o. = 1 roll, 

100 rotl z=z 1 cantaro sottilc, - - =1 74.504 

110 rutl = 1 cantaro groawo. 
MADEIRA I. — Dcnoniinutions and relative values, 
same as Lisbon, Portiujal : 
32 arratol or lib])ra =« 1 arro])a, - - - := 32.34D 

MAUKITlUa I. — Por^Xouis; 10 onces = 1 livro iA)6 

MEXICO. — Standard same as Cadiz, Spain. 

MOROCCO. — 100 rotl = 1 cantaro, - - = 118.723 

Tanr/icrs : 1 rotl (mmrr^'), - - - - =a 1.00 

1 rotl (//jcrrAc/), - - - - - = 1.701 

MOZ AMBlCi U E (Africa) . — Mozambique : Same aa Port- 

Vf/al. 
NORWAY. — Same as Denmark. 

PERSIA. — Bushirc: 3 cheki = 1 ratol, 74 r. 1 maund 
tabree, 2 m. tabrce = 1 maund sliow. 
1 maund show, ba/jir, - - - - =12.5 

1 ** copra, '* - - - - = 7.3 

1 '* j?1k)W, factory, - - - = 13.5 

1 " copra, ** - - - - = 7.75 

Tauris: 2 mascais = 1 dirhem, 50 d. = 1 ratel, 6 

ratel = 1 batman, - - - - = 5.047 

Shiraz: I Initixiixn, - - - - - = 10.125 

PORTUGAL. — 570 grao or 24 cecropulo or 8 outuava, 
= 1 onca, 10 o. = 4 quarta = 2 marca = 1 arra- 
tel, - - - - - - = 1.010 

32 arratol = 1 arroba, 4 arrobe = 1 quintal, - = 130.06 

13.J quintale = 1 tonelada. 
PRUSSIA (Icffal sijicc 1820) : 4 quentchcn = 1 loth, 

2 1. = 1 unze, 8 u. = 1 mark, 2 m. 1 pfund, = 1.0312 
IG.i pfund = 1 liespfund, li 1- = 1 stein, - - = 22.087 
5 stein = 1 centner, 3 c. = 1 schilfpfund, - = 340.31 



FOREIGN WEIGHTS JIEDUCJED TO XJNITED STATES. a 31 

Foreign. U. States, 

Avoirdupoia 
pounds. 

100 pfunds = 1 lagel,/(7r steel, - - - := 103.116 

1 Prussian mark === 1 Cologne mark. 

Da7i^;:^ic; 33 pfund=« 1 stein,/or/aa?, - - = 34.029 

RUSSIA {legal throughout the Empire) : 

3 zolotnik = 1 loth, 32 loth or 12 lana =* 1 font, 

40 funt = 1 pud, - - - - = 36.067 

10 pud = 1 berkowitz, 3 b. =» 1 paken, - =1082.02 

2 paken = 1 last. 

Libau: 20 funt = 1 liespfunt, 20 1. == 1 schiffpfunt, = 364.168 

Hamburg weights are also used here. 
Riga : 20 pfunde = 1 liespfund, 5 1. = 1 lof, - = 92.158 

4 lof =3 1 berkowitz or schiffpfund, - - = 368.633 
SARDINIA I. — Cagliari, &c. : 12 oncia = 1 libbra, 

26 1. = 1 rubbio, 4 r. = 1 cantarello, - - = 93.082 

SICILY I. — iVfe55i7ia; 12 oncia = 1 libbra, ' - = 0.707 

2.i libbre = 1 rotolo ^ottile, - - - = 1.768 

33 oncia = 1 rotolo grosso, - - - = 1.945 

100 rotoli = 1 cantaro {gross or net), 

Palermo : 250 libbre or 100 rotoli sottile =s 1 cantaro 

sottile, - - - - - - = 175.04 

275 libbre or 100 rotoli grosso = 1 cantaro grosso, = 192.556 
Syracuse : 250 libbre or 100 rotoli sottile = 1 cantaro 

sottile, -..-.= 180.125 

275 libbre or 100 rotoli grosso = 1 cantaro grosso, = 198.137 
SPAIN. — 16 Castilian onze = 1 Castilian libra {Cus- 
toms), - - - - - -= 1.01546 

Alicant : 12 onze = 1 libra menor {minor). 

18 onze = 1 libra mayor {major), - - =s 1.144 

24 1. mayor or 36 1. menor = 1 arroba, - - = 27.456 

4 arrobe = 1 quintal, 2i q. = 1 carga, - = 274.567 

8 carga = 1 tonelada. 

24 Ciistilian libre = 1 arroba,/or vermilion, - = 24.371 

25 Castilian libre = 1 arroba of the Customs, - = 25.386 
Barcelona: 25 libre == 1 arroba, - - -=22.14 
Bilboa: 25 libre = 1 arroba, - - - = 26.97 
Cadiz {Standard of Castile) : 8 onza = 1 marco, 

2 m. = 1 libra, 25 1. = 1 arroba, 4 a. = 1 quintal, = 101.546 
20 quintale = tonelada. 
Corunna, Ferrol: 16 onze = 1 libra sutil, 100 libre 

sutil=: 1 quintal (Cfl'5??7/V7?i), - - -=101.546 

20 onze = 1 libra gallega, 100 1. g. = 1 quintal, - = 126.933 

25 libre = 1 arroba. 

Gibraltar : 16 onze = llihT,x {Casiilia?i), - =1.01546 

16 ounces = 1 pound, 25 p. = 1 arroba, - - = 25. 



32 a roEEiGN weights eeduced to united states. 

Foreign, U. Slates, 

AvoirdapoLa 
pounds. 

Malaga. — Same as Cadiz. 

1| quintal, or 34 barrile = 1 carga of raisins, - = 177.7 
Santander: 100 libre = 1 quintal, - - = 152.28 

Valencia : 12 onzc = 1 libreta or libra menor, - = 0.784 

18 onzc = 1 libra gruesa, - - - = 1.170 

35 libre menor =1 arroba, 4 a. = 1 quintal, 

12i arobe = 1 carga, -• - - - = 338.413 

SWEDES. — Stockholm, &c. : 

Viktualic-wigt or skal-wigt : 4 quintin = 1 lod, 2 1. 

= 1 untz, 10 u. = 1 skalpuuJ, - - = 0.9375 

20 8kal[)und = 1 lispund, 20 1. = 1 skoppund, - = 375. 

32 skalpund = 1 stcn, - - - = 30. 

12 «k('ppund = 1 last. 
Metall-wigt or jcrn-wigt {for iron, steel, ttc.) : 
20 mark = 1 markpund, 20 m. = 1 lispund, 

20 1. = 1 skoppund, - - - - = 300. 

15 fcjkcj)pund = 1 last. 

H\ lispund = 1 waag or vog, /or /in, - - = 123.75 

Uppstadt-wigt {inland weight \ : 

400 pund or 20 lispund = 1 skoppun«l, - - = 315.074 

Ta(;}ij<'rn-wigt : 400 pund or 20 lisp. = 1 sk«'ppund, = 453.47 
Bcrg-wigt : 400 pund or 20 lisp. = 1 skoppund, = 348.822 
SWITZKKLAXI) (legal, since 182^^, for the Cantons of 
Aarau, Basle, Ihrne, Freiburg, Lucerne, Solothum, 
Vaud ; l)ut not in general use) : 
8 gros = 1 unze, 8 u. = 1 mark, 2 m. = 1 livrc or 

pfund = 1 livrc poids de marc of France, = 1.07022 

10 livres = 1 stein, 10 s. = 1 «'.'ntnrr. - - = 107.022 

Special and local : 

Berne: 100 pfunde = 1 centner, - - = 114.9 

Geneva : 24 grani = 1 denier, 24 d. = 1 once, 15 o. 

= 1 livre foible, = 1.0118 

18 once = 1 livre fort, - - - - = 1.2141 

Lausanne : 10 onces= 1 livre = 1 livrc poids dc mctrique 

of Franco, = 1.10237 

Neufchatcl: 8 onccs = 1 marc, 2 m. = 1 livre, - = 1.140S2 
St. Gall : 10 unze = 1 loth, 2 1. = 1 pfund, - = 1.2014 

Zurich: 18 unze = 1 pfund, - - - = 1.1037 

Poids foible {for silks ^ &c.) : 2 lothe = 1 unze, 8 u. 

= 1 mark, 2 m. = 1 pfund, - - - = 1.0344 

TRIPOLI. — (N. Africa) : 8 termini = 1 usano, 10 u. 

= 1 rotolo, 100 rotoli = 1 cantaro, - - = 111.214 

400 drachmi = 1 oke, - - - = 2.7429 



FOREIGN WmGHTS REDUCED TO UNITED STATES. a 33 

Foreign, U. States, 

Avoirdupois 
pounds. 

TUNIS. — 8 metical = 1 usano, 16 u. = 1 rotolo, 100 

rotoli = 1 cantaro, - - - = 109.155 

TWiKEY,— Aleppo : 266| meticals = 1 oke, - - = 2.81349 

480 meticals = 1 rotolo, 5 r. = 1 vesno, 20 v. = 1 

cantaro, - - - - - = 506.428 

3} rotoli = 1 batman, lOJ b. = 1 cola, - - = 177.249 

30i rotoli = 1 cantaro zurlo, - - - = 154.46 

400 meticals = 1 rotolo for Damacene, - - = 4.22023 

453^ meticals = 1 rotolo for Persian silks, - = 4.78293 
4665 meticals = 1 rotolo Tripolitan, - - = 4.92361 

Bagdad: 24 vakia = 1 oke, - - - = 2.74286 

Bussorah: 100 miscals = 1 cheko, - - -= 1.02857 

24 vakia = 1 maund, - - - - = IIG. 

46 oke = 1 cuttra, - - - - = 136.482 

24 vakia attaree = 1 maund attaree, - - = 28. 

76 vakia attaree = 1 maund sessee, • - = 88.6666 

4f vakia attaree = 1 vakia. 
Constantinople: 16 kara, kilot or taim = 1 dirhem, 
100 dirhem or 66| meticals = 1 chcki or yusdrum, 
2 cheki = 1 rottel, 100 r. = 1 cantaro, - = 140.3 

13fx rottel = 1 batman, 7i b. = 1 ftmtaro. 
266| meticals or 2 rottel = 1 oka. 

116| meticals = 1 cheki, ybr opium, - - = 1.7578 

Damascus : 400 meticals or 60 peso = 1 rotolo, 100 

rotoli = 1 cantaro, - - - - = 395.673 

Smyrna : 100 drachmi or 66| miscals = 1 cheko, 2^ 

cheki = 1 cequi, - - - - - = 1.7578 

180 drachmi or 120 miscals = 1 rotolo, 13j r. = 1 

batman, 74 b. or 100 rotoli = 1 cantaro, - = 126.571 
4 cheki = 1 oka, 45 0. = 1 cantaro. 
44 oke = 1 cantaro, /or tin, &c., - - - = 123.758 

West Indies, 

In the Islands of Antigua, The ^ Bahamas, Barhadoes, 
Barbuda, Dominica, Grejiada, Jatnaica, Les 
Saints, Montserrat, Nevis, St. Kitts, St, Vincent, ■ 
Tobago, Tor tola, the commercial vreights are the 
same as in Great Britain. 

In Deseade, Guadeloupe, Mariegalante, Martinique, St, 
Lucia : 2 quartiers = 1 marc, 2 m. = 1 livre, 100 

1. = 1 quintal, = 107.922 

3 quintals = 1 charge, 3i c. = 1 millier. 

Thiis being the old system,* poids de marc, of France. 



Ma FOREIGN WEIGHTS REDUCED TO UNITED STATES. 

Foreign. U. States. 

AyoirdupoiB 
pounds. 
In Saba, St. Eustatia, St. Martin, the commercial weights 

are the same as in li(jlUind, old system. 
In Santa Cruz, St. John, St. Thomas, Same aa Denmark. 
In St. Bartholomew , Same as Sweden. 
In Curacoa, Trinidad, Same as Castile {Spain), 
In Bonaire: 100 pond = 1 centenaar, - - - = 103.659 

3 centenaar = 1 schippond. 
This being the old weight of Brabant, Holland. 
Cuba I. — Cardenas, Cienfueyos, Havana, Malanzas, 
Nuevitas, Porto Principe, St, Jago, &c. : Same as 
Castile (Spain). 
IIayti I. — AiLT Cai/cs, Cape Haytien, Jcremic, Port au 
Prince, Port Platte, &c. : Same as Franco. ]>efQro 
1812. 
Savamia, St, Domingo, &c. : Same as Castile {Sftun), 
Porto Rico I. — Ponce, St, Johns, &c. : Same as Castilb. 



35 a 



FOREIGN LIQUID MEASURES REDUCED TO UNI- 
TED STATES. 



Foreign, 



ABYSSINIA. - 



ALGIERS. — 16| litres 



Massuah : 1 cuba, 

1 khoulle, G k. = 1 hectoli 
tre, 

ARABIA. —Mocha: 20 vacias {weight) = 1 niisfiah, 8 
n. = 1 ciida or gudda= 16 lbs. Av., or of oil., &c. 
AUSTRIA {legal and general for the Empire) : 
Vienna., Trieste, Lintz, Prague, Pesth, &c. : 
4 seidel = 1 mass, 10 mass = 1 vicrtcl, 
4 viertel = 1 eimcr or orna, - . . 

32 eimer = 1 fuder, . - - - 

Special and local — 

Trieste : 43 caffiso = 1 orna, ^ - - 

BouEMiA. — 32 pinte = 1 eimer, - - - 

4 eimer = 1 fass, ----- 

Prague: GO mass = 1 eimer, - - - 

Hungary. = 4 rimpel = 1 halbo or iczo, 100 ieze = 1 
czcber, ------ 

1 anthal = 13.352 gals. 1 ako, 
Buda, Pesth, &c. : 1 eimer, - - - - 

Presburg : 1 eimer, - - -• - 

Moravia. — 40 mass = 4 viertel = 1 eimer, 
AZORE ISLANDS. — Same as Lisbon {Portugal), 
BALEARIC ISLES.— Majorca. — 4 quarta = 1 quartes, 
G quartes = 1 cuartin, 4 cuartin = 1 carga, 
3 carga = 1 pelexo. 1 quartinello, 
Minorca. — 4 quarta = 1 quartes, 3 quartes = 1 gerrah, 
10 g. = 1 carga, 4 c. = 1 botta, 

1 barrel, ----- 
BELGIUM. — 3i canette = 1 uper, 10 u. = 1 emmer, 

3 e. = 1 vat = 1 hectolitre of France, 

2 pinte = 1 pot or mingle, 2 p. = 1 stoop or gelte, 
2 s. = 1 schreef, 25 schreef = 1 aam, 

6 J aam = 1 ton of spirits, for shipping, 

2| stoop = 1 velte, - - - - 



U, States 

Wine 
gallons 

:= 0.2G8 
= 2G.418 
= 2.07 



3.73S 
14.952 

478.48 

14.952 
1G.141 
G4.5G 
1G.591 

22. 

1S*.495 
15.028 
19.3G8 
11.282 



28.066 
1.8 

133.379 
8.344 

-= 26.418 

36.578 
237.76 
2.011 



36 a FOREIGN LIQUID MEASURES REDUCED TO UNITED STATES. 

Foreign, U. States, 

Wine 
gallons. 

BERMUDAS ISLANDS. — Same a^ United States. 

BRAZIL. — Bahia : 10 garrafa = 1 Canada, - = 18.734 

10 Canada = 1 pipa of molasses, - - -= 187.342 

7i Canada = 1 pipa of spirits, - - - = 134^880 
Rio Janeiro : 4 quartilho = 1 medida, 3 m. = 1 pote, = 2.185 

16 pote = 1 pipa, 2. p. = 1 tonelada, - - = 262.178 

1 fiasco, .... . = 0.562 

Centi^al and South America, 

Balize, Guatimala, Yucatan, Bolivia, Buenos Ayres, 
Equador, New Granada, Paraguay, Peru, Uru- 
guay, Venezuela. Denominations and values, 
same as Castile (Spain.) 
Guiana. — Bcrbicc, Denier ar a , Essequibo, Surina^n : Same 
as Holland. 
Cayenne. — Same as France. 
CANARY ISLANDS. — Grand Canary, Teneriffe, &c. : 

4 cuartilla = 1 arroba, 28i a. = 1 pipa, - = 120.06 

CANDIA I. —By locight — 1 oke = 2.04U lbs. Av. 
CAPE COLONY. — Cape Town: 16 flask = 1 anker, 

4 anker = 1 aam, 4 aam = 1 legger, - - = 152. 

Cape VERDE I. — Same as Lisbon {Portugal). 
CHILI. — Valparaiso, Coquimbo, &c. : 4 copa = 1 quar- 

tilla, 4 q. = 1 arroba, - - - = 9.906 

CIHNA. — By weight. See Weight^. Also — 

10 kop tsong = 1 shing tsong, 10 shing tsong = 1 

tau tsong, 5 tau tsong = 1 liok tsong, 2 hok tsong 

= 1 shik tsong. If shik tsong = 1 yu, 5 yu = 1 

ping = 832 lbs..Av. 

CORSICA I. — 4 cuarto = 1 boccale, 9 b. = 1 zucca, 

12 z. = 1 barile, - - - - = 36.985 

CYPRUS I. — By weight. See Weights. 
DENMARK. — - 155 pagel or 38} potte or 191 kande = 

1 anker, .-.-.= 9.889 

4 anker = 1 aam, IJ a. = 1 oxehoved, 4 o. = 1 

fader, li f. = 1 stykfad, • - - - = 296.672' 

60 viertel = 1 piba, --.-== 122.499 
17^ viertel = 1 toende,/(;r beer, - - - = 34.708 

EGYPT. — By weight. See Weights. 
FRANCE. — 1000 millilitres == 100 centilitres = 10 de- 
cilitres = 1 litre, - - - . = 0.264 
100 litres = 10 decalitres = 1 hectolitre, - - = 26.418 
100 hectolitres = 10 kilolitres = 1 myrialitre. 



FOREIGN LIQUID MEASURES REDUCED TO UNITED STATES, fl 37 

Foreign, U. States, 

Wine 
gallons. 

GEEMxiNY. — Baden (legal) : 4 schoppen = 1 mass, 
12i m. == 1 stutz, 8 s. = 1 ohm == I4 hectolitres of 

France, = 39.626 

10 ohm = 1 fuder. 
Manheim : 16 schoppen or 4 eich-mass = 1 viertel, 12 

viertel = 1 ohm, - -^ - - = 25.285 

16 schoppen or 4 wirths-mass == 1 viertel, 24 viertel 
= 1 ohm = 16 decalitres of France, - - = 42.268 

Bayaria {legal) ; 4 quart 11 = 1 mas or masskanne, 60 

masskanne = 1 eimer, - - - - = 16.944 

64 ** =1 eimer, /or beer, - - = 18.075 

Augsburg J Wurtzburg : 4 achtel = 1 seidel, 16 s. = 1 

beson, 8 beson = 1 eimer, 12 e. = 1 fader, - = 238.745 
Nuremberg: visir-mass : 4 seidel = 1 viertel, 32 v. 

= 1 eimer, 12 e. = 1 fuder, - - - = 232.348 

Schenk-mass : 32 viertel = 1 eimer, - - = 18.244 

12 eimer = 1 fuder, - - - - = 218.924 

Ratisbon : 32 viertel == 1 eimer, 12 e. = 1 fuder, - = 158.507 

Hanover (legal) : 8 nossel = 4 quartier = 2 kanne = 1 

stubchen, - - - - - = 1.036 

10 stubchen = 5 viertel = 1 anker, - - = 10.36 

4 anker or 2^ eimer = 1 ahm, - - - = 41.439 

6 ahm or 4 oxhoft = 1 fuder, - - - = 248.637 

4 ahm or tonne, for beer, = 1 fass, - - = 165.756 
Bremen : 180 mingel == 90 versel = 45 quartier or 

vierling = 11^ stubchen = 5 viertel = 1 anker, = 9.574 
24 anker == 6 ohm = 1 fuder, - - = 229.779 

16 mingles = 1 stubchen, 6 s. = 1 stechkanne, 6 

stechkanne = 1 tonne, /or whale oil, - - = 30.64 

44 stubchen (beer ineasure) = 1 tonne, - = 43.836 

Hesse Cassel : 4 schoppen == 1 mass, 4 m. = 1 viertel 

or quartlein, 20 v. = 1 ohm, 6 ohm = 1 fuder, = 251.547 
20 viertel (beer measure) = 1 ohm, - - = 46.128 

Hesse Darmstadt (legal) : Denominations and relative 
values, same as H. Cassel. 
1 ohm = 16 decalitres of France, - - = 42.27 

Frankfort: 4 schoppen = 1 mass, 44 neu-mass = 1 

viertel, 20 v. = 1 ohm, 6 ohm = 1 fuder, - = 227.352 

8 alt-mass = 9 neu-mass. 

1^ fuder = 1 stlickfass, .. - - - = 303.136 

HoLSTErx. — Hamburg, Altona, Lubec : 8 oessel, plank or 

nossel = 4 quartier = 2 kanne = 1 stubchen, . = 0.955 
8 stubchen = 4 viertel = 1 eimer, - - - = 7.644 

5 eimer or 4 anker = 1 ahm or fass, - - =38.22 



38 a FOREIGN LIQUID MEASURES REDUCED TO UNITED STATES* 

Foreign, U. Slates. 

Wiiic 
galloiig. 

Ij ahm or li tonne = 1 oxhoft, - - - = 57.33 

4 oxhoft or 2 pipe = 1 fuder, - - - = 229.32 

16 margel or melgel = 1 stechkanne, G 8. = 1 tonne, 

2 t. = 1 quarteel,/or whale oil, - - - = G1.548 

1 tonne, /or beer, -_-.=: 45.804 
Mecklenburg. — Rostock, &c. : Same as IIolstein. 

Saxony. — Dresden : 8 quarticr or 2 nossel = 1 kannc, 

quart or ehenkkanne, 3 kanne = 1 viurtel, - = 0.743 

18 V. = 1 ankcT, Ij a. = 1 eimer, - - =: 17.H33 

2 e. = 1 anker, Ij a. = 1 oxhoft, - - - = 53.499 
15 o. = 1 fass, 22 f. = 1 fuder, - - = 213.990 
4 tonne or 2 viertel = 1 fans, beer measure, - - = 104. 02G 

Leipsic : Denominations and rehitive values same as 
Dresden, but eapaeity values = 12.53 ^Iq greater. 
24 viertel = 1 eiiiier, - - - - = 20.067 

Frcyburg : 100 sehoppen or viertel = 1 brente, - = 10.312 

10 In'onte = 1 fans, - . - - = 1G5. • 

GREAT BRITAIN : Imperial measure : Denominations 

and relative values same as icine measure, U.S., 

but capacity values 20 fV^ P^^ cent, greater. See 

Liquid Measures, U. S. 

GREECE. — Patras : 24 bocealc = 1 barile, - - = 13.54 

HOLLAND {l((jal) : 10 vingerhoed = 1 maatje, 10 m. 

= 1 kan, 10 k. = 1 vat = 1 hectolitre of F., = 26.418 

Previous to 1820 — 

2 mutsje = 1 pint, 2 p. = 1 mingle, 2 m. = 1 stoop, 
32^ s. = 1 viertel, 2^ v. 1 steekan, 2 s. = 1 anker, 

4 a. = 1 aam, li aaiu = 1 okshoofd, - - == 61. 

1 aam, yor oil, - - - - - = 37.64 

1 legger, /or Z>ctr, - - . . = 153.57 

India a?id Malaysia or East Lidies. 

Ceylon I. — Colombo : 4 aams = 1 Icgger, - - = 150. 

IIiNDOSTAN. — Bombay: GO rupees = 1 seer, 50 seers = 
1 maund fluid = 77 lbs. A v. 
Calcutta: 5 sicca = 1 chattac, 4 c. = 1 pouah, 4 p. 
= 1 seer, 5 s. = 1 pussaree, 8 p. = 1 bazar 
maund weight = 82.13 lbs. Av. 
Madras : By weight. See Weights. 
Scrampore, Tranquebar, {legal) : Same as Denmark. 
Java I. — Batavia : 5. kiin = I harile, - - = 13.207 

Luzon I. — Manilla: Same as Cadiz {Spain). 



FOUEIGN LIQUID MEASURES REDUCED TO UNITED STATES. (2 39 

Foreign. U, States. 

Wine 
gallons. 
SiAM. — Bangkok : 20 canan = 1 cohi = 5 decalitres of 

France, - . . . .= 13.209 

Sumatra I. — 8 pakha or 2 culah = 1 koolah, 15 koolah 

= ltub, = 17.44 

lONLlN ISLANDS. — Cephalonia : 2 quartucci = 1 
boccale, 8 b. = 1 pagliazza, IJ p. = 1 secchio, 6 

8. = 1 barile, - - - - - = 18. 

Corfu ^ Paxos : 1^ miltre = 1 boccale, 12 b. = 1 sec- 
chio, 6 8. = 1 barile, - - - = 18. 

Zante : IG quartucci or 8 boccale = 1 lira, IJ 1. = 1 

secchio, G s. = 1 barile, - - - - = 18. 

ITALY. — Lombard Y and Venice. — Government and 
Customs measure: 10 coppi = 1 pinta, 10 p. = 1 
mina, 10 m. = 1 soma = 1 hectolitre of France, =*= 26.418 
Special and local — 
Venice: IJ quartuzzi = 1 boccale, 2| b. = 1 bozza, 4 

bozzi = 1 secchio, G s. = 1 mastello or concia = 17.119 
2 mastelli = 1 bigoncia, 4 b. = 1 anfora, - = 13G.05 
li anfori = 1 botta, - - - - = 171.187 

IG miri = 1 bigoncia, 24 b. = 1 migliajo, 2 m. = 

1 botta, /or 027, - - '- - =322.22 

Naples : GO carraffa = 1 barile, - - - = 11.581 

3i barile = 1 salma, 4 s. = 1 pipa, - - = 1G2.137 

12 barile = 1 botta, 2 botte = 1 carro. 
G misurella or 1\ pignata = 1 quarto, 16 q. = 1 

stajo, 16 s. = 1 salma, /or oil, - - = 42.538 

11 salma = 1 last for shipping. 
Sardinia. — Genoa: 90 amola or 5 foglietta or pinta = 

1 barile, 2 b. = 1 mezzaruola, - - - = 39.218 

16 quarteroni = 1 quarto, 4 q. = 1 barile, /or oil, = 17.084 
Turin: 20 quartine = 1 boccale, 2 b. = 1 pinta, 6 p. 

= 1 rubbio, 6 r. = 1 brenta, 10 b. = 1 carro, = 148.806 
States of the Church. — Ancona: 

4 fogliette = 1 boccale, 24 b. = 1 barile, - = 11.35 

1 soma of oil, - - - - - = 18.494 

Rome: Wine measure: 4 cartocci = 1 quartuccio, 4 

q. = 1 foglietta, 4 f . = 1 boccale, - - =0.481 

32 boccali = 1 barile, 16 b. = 1 botta, - - = 246.544 

Oil measure: 4 boccali = 1 cugnatello, 10 c. = 1 

mastello or pello, 2 m. = 1 soma, - - = 43.333 

Tuscany. — Leghorn, Florence, Pisa : 

4 quartucci or 2 mezette = 1 boccale, 40 b. or 20 
liasco = 1 barile = 133J libbra or 99.81 lbs. Av. 



40 a FOREIGN LIQUID MEASUEE3 REDUCED TO UNITED STATES. 

Foreign. U. States. 

Wine 
gallons. 

16 fiasco = 1 barile, 2 b. = 1 botta, for oil = 240 
libbra or 179| lbs. U. S. 
MALTA I. — 2 caffisi = 1 barUe = 50 rotl or 87i lbs. 

Av. 
MADEIRA I. — Standard same as Lisbon {Portugal). 

MAURITIUS I. — Port Louis : 1 velt. - - = 2.^ 

MEXICO. — Same as Cadiz {Spain). 

NORWAY. — Same as Denmark. 

PORTUGAL. — Lisbon, &c. : 24 quartilhi or 6 Canada 

= 1 alquoire or cantaro, - - - - = 2.185 

2 alqueirc = 1 almudc, 20 a. = 1 bota or pipa, = 113.027 
2 bota = 1 tonclada. 
31 almudes = 1 pipa, London gauge. 
Oporto : 2 al((ueirL' = 1 almude, 21 a. = 1 pipa, = 139.134 

PRUSSIA {Irgal througJwut the kingdom since 1820 :J 

2 ossel = 1 quart, 30 q. = 1 anker, 2 a. = 1 eimcr 
= 3840 cubic Rhein-zolle or 4192 cubic inches, 

U.S., .... = 18.146 

3 eimer or 14 ohm = 1 oxhoft, 4 o. = 1 fudcr, - = 217.758 

3 J eimcr = 1 fass. 

Dantzic: 3^ eimer = 1 fass, - - - = 00.487 

eimer = 1 pipe, - - - - - = 108.876 
Konigshcrg : 44 eimer = 1 pipe, - - - = 81.058 

RUSSIA {legal for the Empire since 1820) : 

124 tseharka = 1 oj^muschka or krawlika, 2 o. = 1 

tschet-werk, 4 t. = 1 vedro or wedro, - - = 3.240 

40 vedro = 1 botschka or anker, - - =; 129.80 

13^ botschka = 1 sarokowaja. 
Revel, Piga: 30 stof = 5 viertel = 1 anker, - - = 10.311 

4 anker = 1 ahm, a. = 1 fuder, - - = 247.46 
SARDINIA I. — 4 quartucci = 1 quartaro, 8 q. = 1 

barile, - - - - - -= 8.874 

2 barile = 1 mezzaruola. 
SICILY I. — Palermo : 20 quartucci = 1 quartaro, 8 

quartari = 1 Imrile, - - - - = 9.430 

12 barile or 5 salma = 1 botta or pipa, - - = 113.237 

12 salma = 1 tonnellata. 

1 cafBso,/(>r oil= 124 rotoli grosso, - - = 3.09 
Messina : 12 barile or 5 salma = 1 pipa, - - = 108. 

1 cafliso, /'or oil = 124 rotoli grosso. 

Syracuse: 12 salma = 1 tonnellata, - - - = 247. 

SPAIN. — For Customs values, see Cadiz. 

Alicant: 4 copa = 1 cuartilla, 4 c. = 1 cuarto, 4 
cuarti = 1 cant arc := 1 Castilian arroba = 25.38 



FOREIGN LIQUID MEASURES REDUCED TO UNITED STATES, ail 



Foreign. 

lbs. = 3.04 gallonsof wine or 3.642 gallons of 90 
per cent, alcohol, United States measure. 
40 arrobe = 1 pi pa, 2 p. = 1 tonelada. 
Barcelona: 4 petricon = 1 mitadella or porrone, 4 m. 
= 1 quartera, 2 q. = 1 cortan or mitjera, 2 c. = 
1 mallah, 8 m. = 1 carga = 12 arrobe or 265.08 
lbs. Av. 
4 carga = 1 pipa. 

4 quarta = 1 cuarto, 4 c. = 1 cortan, 30 cortan = 
1 carga, /or oil = 11 arrobe or 243.54 lbs. Av. 
Cadiz {Standard o/ Castile) : 2 copa = 1 azumbra, 2 
a. = 1 cuartilla, 4 c. = 1 cantaro or arroba. 
1 arroba mayor = 35 liljre or 35.541 lbs. Av., or 

984| cubic inches of distilled water at 60^, 
1 arroba menor, /or oil = 27i libre, 
16 arrobe = 1 mayo, 27 arrobe = 1 pipa, 30 arrobe 
= 1 bota, 2 bote = 1 tonelada. 
Corunna, Ferrol : 16 quartilli = 1 olla, 4 o. = 1 can- 
ado, 4 c. = 1 mayo = 14 arroba sutil, - 
Gibraltar : Same as tlie United States. 
Mala/ja: 8 azumbre = 1 cantaro or arroba (34 J 1.), 
Santander : 8 azuml)re = 1 cantaro = 26 libre, - 
Valencia: 1 arroba = 38 libre menor, 

1 arroba, /or oil= 20-g- libre menor, 
SWEDEN". — Stockholm, &c. : 4 jungfru or ort = 1 

quarter, 4 q. = 1 stop, 2 s. = 1 kanna, - 
6 kanne = 1 atting or ottingar, 2 a. = 1 fjerding, 
1-j f . = 1 ankare, 2 a. = 1 embar. If e. = 1 
tunna, _ _ _ . - 

li tunna = 1 am, l.J a. = 1 oxhufwud, 

2 oxhufwud = 1 pipa, 2 p. = 1 fuder, 
SWITZERLAND (legal since 1823, for the Cantons of 

Aaraii, Basle, Berne, Freiburg, Lucerne, Solothurn, 

Vaud ; but not in general use) : 
10 emine = 1 mass or pot, 10 m. = 1 gelt = 1^^ 

decalitres of France, . . _ _ 

Special a7id local — 
Basle : 4 schoppen = 1 mass, 4 m. = 1 viertel, 2f 

= 1 setier, 1| s. = 1 ohm, 3 o. = 1 saum, - 
Berne: 8 becher or 2 viertel = 1 mass, 25 m. = 

brente or eimer, 4 b. = 1 saum, - 
4 saum = 1 fass, 14 f. = 1 landfass, - 
Geneva : 2 pot = 1 quarteron, 24 q. = 1 setier. 



U. States. 

Wine 
gallons. 



4.263 
3.319 



-= 42.533 

4.182 
4.739 
3.566 

2.737 
-= 0.689 



33.174 
62.202 
248.81 



= 3.5664 



= 40 337 

= 44.161 
= 264.966 
= 11.942 



42 a FOREIGN LIQUID MEASURES REDUCED TO UNITED STATES. 

Foreign, U, States, 

Wine 
gallons. 

12 setier = 1 char. 
Lausanne : 10 verre = 1 mass, 10 m. = 1 broc, 3 b. 

= 1 setier or eimer, - - - - = 10.699 

Neufchatel: 8 pote or mass = 1 brochet or stutz, = 4.024 

24 brochet = 1 brande, 2| brande = 1 gerl, - = 26.159 

24 brochet = 1 muid, 2i m. = 1 bosse. 
St, Gall: lautcr-mass : 8 mass = 1 yiertel, - = 2.773 

4 viertel == 1 eimer, 4 e. = 1 saum, - - = 44.37 

7J saum = 1 fudcr. 

8chenk-mass : 8 mass = 1 viertel, - - = 2.403 

4 viertel = 1 eimer, &c. 

8 mass = 1 viertel, /(^r oi/, - - - = 2. 867 

Zurich : lauter-mass : 8 statz or 2 mass = 1 kopf, = 0.1>G37 
74 kopf = 1 viertel, 4 v. = 1 eimer, 14 e. = 1 saum, = 43.368 
Schcnk-mass : 2 quartli = 1 mass, 2 m. = 1 kopf, = 0.86740 
74 kopf = 1 viertel, &c. 
TRIPOLI (N. Africa) : 14 caraffa = 1 mataro, for oil 
= 42 rotoli = 46 iV lbs. Av. 
1 barril = 116J rotoli. 
TUNIS. — 2 mettar, /or xoine = 1 mcttar, for oil = 36 
rotoli = 39 fV lbs. Av. 
1 millerolle = 120 rotoli. 
TURKEY. — lalmud, - - - . = 1.38 

West Indies, 

In the islands of Aniifjua^ the Baha/nas, Barbadocs, Bar- 
buda, Domiiiicay Grenada, Les iSaints, Monlserratf 
Nevis, St. Kitts\ St. Vincent, Tobago, Tor tola, 
the Measures for Liquids are the same as those of 
the United States, or the same as those of Great 
Britain, previous to 1825. 

In Jamaica: 85 Imperial gallons = 1 puncheon, - = 102.03 

In the Islands of Descade, Guadeloupe, Mariegalante, Mar- 
tinique, St. Lucia : 8 muces = 4 roquilles = 2 
chopines = 1 pinte, 8 pintes = 4 pots = 2 gallons 

= lvelt, = 2. 

35 veltes = 1 muid. 
This w^as the system for Liquid Measures in France, 
before 1812, except that the value of the muid 
was 70.855 gallons, U. S. 

In Bonaire, Curacoa, Saba, St. Eustatius, St. Martin: 
Same as Holland old measure, or measure before 
1820. 



FOREIGN LIQUIB 31EASURES REDUCED TO UNITED STATES. a43 



Foreign, 

In St. Bartholomew : Same as Sweden. 

.In Trinidad: Same as Castile (Spain), 

In Santa Cruz, St. John, St. Thomas : Same as Denmark. 

Cuba I. — Cardenas, Cienfuerjos, Matanzas, Nuevitas, 

Porto Principe, St. Jar/o, &c. : Same as Castile 

(Spain). 
Havana: 1 arroha = 4.1 gallons. 1 bocoy, - = 

Hayti I. — Aux Caijes, Cape Haytien, Jeremie, Port au 

Prince, Port Platte, &c. : Same as France before 

1812. 
Savanna, St. Dmingo, &c. : Same as Castile. 
Porto Rico I. — Same as Castile. 



TJ, Slates, 

WLne 
gallona 



36. 



44 a 



FOREIGN DRY MEASURES REDUCED TO UNITED 

STATES. 

Foreign. U. States. 

Winches tor 

ABYvSSTNIA. — Ma^suah : 24 madcga = 1 ardcb, - = (>.:5o3 

ALGIERS. — 2 tarrie = 1 saa or suha, - - - = 1 . 125 

8 sjia = 1 ciifRso, ----:= \), 

100 litres = 1 hectolitre, - - - - = 2.838 

ARABIA. — Mocha: 40 kella = 1 tomaun (for rice) = 
1G8 lbs. Av. 

AUSTRIA {Iffjal and general) : 

2 becher = 1 fudermassel , 2 f . = 1 muhlmassel, 2 
m. = 1 acht«'l, 2 a. = 1 viertel, 4 v. = 1 metze, 

30 m. = 1 math, - - - - = 52.354 
Local and special — 

Trieste: 2 pulunic = 1 metze, li m. = 1 stajo, - = 2.15G 
Bohemia. — Prague : 12 seidel = 1 massel, 4 m. = 1 

viertel, 4 V. = 1 strich, - - - = 2.(»56 

3 viertel = 1 metze, - - - -= \.\)\)2 
Hungary: 32 hal])e = 1 viertel, 4 v. = 1 metze, - = 1.745 

Buda and Pcsth : 1 metze, - - - - = 2.27 

Moravia : 1 metze, - - - - - = 2. 

AZORE I. — 2 meio = 1 alqiieire, 4 a. = 1 fanga, - = 1.35'J 

BALEARIC I. — M.uoRCA : ])arcella = 1 quartern, = 2.042 

Minorca: 6 barcella = 1 quartera, - - = 2.150 
BELGIUM.— 100 uper = 10 setier = 3 mudde = 1 

inuid = 3 /i^r/o//7rc5 of France, - - -= S.51.'> 

10 muid = 1 last, - - - - = 85.134 
Antwerp, Brussels, &c. (old measures) — 

10 malster vat = 1 halster, 10 h. = 1 sac, - = G.918 

108 gelte = 1 muid, - - - - = 8.302 

GAcn^ ; 1 halster, - - - - -= 1.4'I'J 

4 meuke = 1 raziere, - - - - = 2.2<") 
Mechlin: 1 meuke, - - - - -=0.(314 

BERMUDxVS I. — Same as United States. 

BRAZIL. — IG quarta = 1 fanga, 15 f. = 1 moio, - = 23.02 

Bahia: 1 alqueire, - - - - -= 0.8G3 
Rio Janeiro : 1 alquoire, - - - - =1.135 



FOREIGN DRY MEASURES REDUCED TO UNITED STATES, a 45 



Foreign, 



U. States. 

"Winchester 
buahelB. 



Ceiitral and South America. 



Belize, Campeche, Nicaragua, San Salvador, Sisal, &c, 
Buenos Ay res, Callao, Carthagena, Laguayra, Mara- 
caybo, Montevideo, Truxillo, Valparaiso, &c. : Same 
as Cadiz, generally. 
Buenos Ay res : 1 fanega, - - - 

Montevideo : 1 fanega, - - - 

Valparaiso : 1 fanega, , - - - 

Berbice, Dcmerara, Essequibo, Surinam: Same as Hol- 
land before 1820. 
Cayenne : Same as France. 
CANARY ISLANDS. — 12 celemine = 1 fanega, 

17 celemine = 1 fanaga {heaped) . 
CANADA EAST. — 1 minot, 

CANDIA I. — learga, . . . . 

CAPE COLONY. — Cape Town : 4 schepel = 1 muid, 

10 muid = 1 load, - - - - 

CHINA. — By weight. See Weights. 
CORSICA I. — G bacino = 1 mezzino, 2 m. = 1 stajo, 
CYPRESS I. — By weight. See AVeights. 
DENMARK. — 4 scxtingkar or Qcrdingkar = 1 otting- 
kar or skieppe, 2 o. = 1 fjerding or gtubchen, 4 
f . = 1 toende, - - - - - 

22 toende = 1 last, - . - - 

EGYPT. — Alexandria, Rosctta : 1 kisloz, 

24 robi = 1 rebob, - - - - 

Cairo : 24 robi = 1 ardeb, - - - - 

FRANCE. — 100 litres = 10 decalitres = 1 hectoUtre, 

100 hectolitres = 10 kilolitres = 1 myrialitre, 
GERMANY .— Baden {legal) : 1000 bocher or 100 mas- 
sel or masslein or 10 sestcr = 1 malter, 
10 malter = 1 zober = 15 hectolitres of France, 
Manheim, Heidelberg : 32 masschel or 4 immel or invel 
or 2 kumpf or vicrling = 1 simmer, 2 s. = 1 vi- 
crnzel, 8 v. = 1 malter, /or wheat, - 
1 malter, minim, %~ 

1 " for barley and oats, - - - 

Bavaria {legal) : 8 masslein or 4 dreissiger = 1 achtel 
or massel, 4 achtel = 1 viertel, - - - 

12 viertel = 1 scheffcl, - - - - 

144 mctzen or 12 mass = 1 scheffel, /or oats, &c., 
4 kubel = 1 seidel, 6 s. or 4 scheffel = 1 muth, /or 
coals and lime, . - - - - 



3.752 
3.868 
2.572 



= 1.776 

= 1.111 

= 4.323 

= 30.65 

= 4.256 



3.947 
86.836 
4.85 
4.462 
5.165 
2.838 
283.782 

4.256 
42.567 



3.152 

2.922 
3.546 

0.526 
=- 6.31 
= 7.363 

= 25.24 



46 a FOREIGN DRY MEASURES REDUCED TO UNITED STATES. 

Foreign. U, States, 

Winchester 

bushels. 

Bamberg : 40 gaissil = 1 simra, 3 s. = 1 shcffel, = G.018 

Bayrcuth: 16 mass = 1 simmer, - - - = 14.044 
Nuremberg : IG mass or 2 diethaufe = 1 metzc, 10 

metzen = 1 malter or simmer, - - . - = 9.028 
16 hafer-mass = 1 hafer-metze, 32 hafer-metzen = 

1 hafer-simmer, - - - - =16.006 

Ratisbon: 4 massel = 1 strich, - - - = 0.750 

1 strich, ybr salt, &c., - - - - = 1.51 
Wurzberg : 144 massel = 12 mass = 2 achtel = 1 

scheffel, - - - - - -= 5.183 

1 scheffcl,/(;r oats, &c., - - - = 8.533 
Hanover {legal) : 144 krus = 24 vierfus = 18 drittel or 

metzen = 6 himten = 1 malter, - - = 5.296 

8 malter = 1 wispel, 2 w. = 1 last, - - = 84.736 

12 malter = 1 fader, - - - - = 63.552 

Bremen : 16 spint = 4 viertel = 1 schcffel, - - = 2.021 

40 scheffel = 1 last, - - - - = 80.834 

Hesse Cassel : 64 kopfehen = 32 masschen = 8 motzen 

= 4 mass = 2 himten = 1 schellel, - - = 2.28 

3 scheffel or 14 viertel or butte = 1 malter, - = 6.482 
Hesse Darmstadt (legal) : 64 kopfehen = 32 maaschen 

= 8 geseheid = 2 kumpf = 1 metze, - - = 0.452 

2 metzen = 1 simmer, 4 s. = 1 malter, - = 3.632 

4 butte (coal measure) = 1 mass, - - - = 17.736 
Frankfort: 4 sehrott = 1 mius8, 4 m. = 1 geseheid, 4 

g. = 1 sechter, 2 8.= 1 metze, 2 m. = 1 simmer, 

4 8. =: 1 achtel or malter, - - - =» 3.256 
HoLSTEiN. — Hamburg, Altona: 4 masschen = 1 spint, 

4 8. = 1 himt, 2 h. = 1 fass, - - - = 1.495 

20 fass = 10 scheffel = 1 wispel, - - = 29.892 

4.i wispel or 14 last = 1 stock, - - - = 134.514 

10 scheffel (for barley and oats) = 1 wispel, - =: 44.831 

45 tonne (for coals) or 30 sacks = 1 fas8, - - =: 179.6 

Lubec: 4 fass = 1 scheffel, 4 s. = 1 tonne, - = 3.796 

3 tonne = 1 dromt, 8 d. = 1 last, - - - = 91.1 
96 scheffel or 24 tonne = 1 last, /or oats, - = 106.918 

Kiel: 4 scheffel = 1 tonne or barril, - • - - = 3.367 
Mecklenburg. — Rostock, Sec, : 16 spint = 4 fass = 1 

scheffel, 2 s. = 1 dromt, - - - = 13.243 

2§ dromt = 1 wispel, 3 w. = 1 last, - - = 105.944 

45 viertel or 15 stubchen = 1 dromt, /or oats, - = 14.9 
Saxony. — Dresden, Leipsic : 4 masschen = 1 metze, 4 

m. = 1 viertel, 4 v. = 1 scheffel, - - = 2.963 

12 s. = 1 malter, 2 m. = 1 wispel, - - = 71.11 



FOREIGN DRY MEASURES REDUCED TO UNITED STATES, fl 47 
Foreign, U. States, 

Winchester 
bushels. 

GREAT BRITAIN : Imperial measure : See Dry Meas- 
ures of the United States ; 1 bushel, - -= 1.0315 

GREECE. — - Pairas : 2 medimni = 1 staro, - = 2.23 

Ibachel, - - -= 0.85 

HOLLAND (legal) : 10 maatje = 1 kop, 10 k. = 1 
scheppel, 10 s. = 1 mudde or zac, 30 m. = 1 last 
= 3 kilolitres of Franco, - - - = 85.134 

India and Malaysia or East Indies. 

BiRMAH. — Rangoon : 2 lamyet = 1 lame, 2 1. = 1 sale, 
4 8. = 1 pyis, 2 p. = 1 sarot, 2 s. = 1 salt, 4 
salt = 1 ten or basket = 10 vis = 58.4 lbs. Av. 
Ceylon I. — Colombo : 24 seers = 1 parah, - - = 0.721 

HiNDOSTAN. — Bombay : Salt measure — 

lOi adowlies = 1 parah, 100 p. = 1 anna = 93.033 

cub. feet, 16 anna = 1 rash, - - =1196.13 

Grain measure — 2 tipprees = 1 seer, 4 s. = 1 adoulie, 
IG a. or 7 pallie = 1 para = 18G| lbs. Av. 
8 para = 1 candy. 
Calcutta : 5 chattac = 1 khoonka, IG k. =1 raik, 4 r. 
= 1 pallie, 20 p. = 1 soallie = 154 lbs. Av. 
8 soallie = 1 morah or maund bazar. 
IG morah = 1 kahoon. 
IJ bazar maunds = 1 soallie. 
Madras : 8 oUock = 1 puddy, 8 p. = 1 marcal, 5 m. 

= 1 para, - - - - - = 1.744 

80 para = 1 garce, - - - - - = 139.535 

Scrampore, Tranquebar {legal) : Same as Denmark. 
Tatta : 4 puttocs = 1 twier, 4 t. = 1 cossa, 60 c. = 
1 carvel = 54 Tatta maunds or 408| lbs. Av. 
Java I. — Batavia: 22 mndden = I coy ang, - = 62.432 

Bantam : 1600 bambou = 400 gantang = 52 mudde 

= 1 coyang,/or Wcc, - - - -=147.565 

Luzon I. — Manilla: Same as Cadiz (Spain), 
Malacca. — 32 mudde = 1 coyang, - - - = 90.81 

SiAM. — 40 sat = 1 sesti, 40 s. = 1 cohi, 10 c. = 1 co- 
yang = 32 Acc^o/zYres of France, - - -= 90.81 
Sumatra I. — 4 pakha = 1 culah, 2 c. = 1 koolah, 15 

koolah = 1 tub, - - - - = 1.872 

IONIAN ISLANDS. — Cor/w, Paxos : 2 misura = 1 

bacile, 4 b. = 1 moggio, - - - - = 4.777 

Cephalonia: 8 misure = 4 bacile = 1 moggio, - = 5.6 
Zante: 8 misure or 4 bacile = 1 moggio, - - = 5.116 



48 a FOEEIGN DRY MEASURES REDUCED TO UNITED STATB5, 

Foreign. IL Stales » 

Winchester 
buteheU. 

Ithaca: 5 bacile = 1 moggio, - - - -= 5. 

ITALY. — LoMBARDY AND Venice. — Govcmment and 
Customs Measure: 10 coppi = 1 pinta, 10 p. = 

1 mina, 10 m. = 1 soma = 1 hectolitre of France, = 2.838 
Special and local — 

Venice : 4 quartaroli = 1 quarto, 4 q. = 1 stajo or 

Btaro, 4 8. = 1 moggio, - - - =s 9.08 

Naples. — 3 misura = 1 stopcllo, 4 s. = 1 mezetta, 2 m. 

= 1 tomolo, 3G t. = 1 carro, - - - == 54.81 

Sardinia. — Genoa: 12 gombctti = 1 ottaro or quarto, 

8 quarto == 1 mina, - - - - = 3. 420 

8 mine = 1 mondino.y/^r salt. 
Nice: 4 motureau = 1 quarticr, 4 q. = 1 Ftajo, 4 stiiji 

= 1 sacco, - - - - - = 3.405 

Turin : 20 cucchiari == 1 copello, 4 c. = 1 ouarticrc, 

2 q. =s 1 mina, 2 m. = 1 ntajo, 3 8. = 1 sacco, *= 3.263 
States of the Cnrucii. — Ancona : 

4 provonda = 1 coppaor lappa, 2 c. = 1 corl^a, = 2.03 
Rome: 54 quartucei or !§ S(M)rzi = 1 starello, 2 8tarclli 

or li istaji = I ((uartarcllo, - - -= 1.044 

4 quartarelli or 2 (iuarto = 1 rubbiatilla, 2 rubbia- 

tille = 1 rubhio, ----== 8.35G 
41 rubbi = 1 tunncllata {^hippinj). 
TuscAi^Y. r— Florence J Lff/horn, Pisa: 

8 bussole = 4 quartucci = 2 mczcttc = 1 mctadella, 
4 m. = 1 quarto, 4 q. or 2 mina = 1 stiijo, 3 8. 
= 1 sacco, 8 sacci = 1 maggio, - - - = 1G.502 

JAPAN.— 10 gantang = 1 ick..ga, 100 ickoga = 1 

icmagoga, 100 icmagoga = 1 managnga. 
MALTxV. — 1 sahna (rasa), - -' - = 8.22 

1 sahna (colnia), - - - -= 0.5G 

MADEIRA I. — Standard same as Lisbon {Portugal), 
MEXICO. — Same as Cadiz (S})m7i). 

MOliOCCO. — Mof/adorc: limid, - - - = 5.184 

NORWAY. — Same as Denmark. 
PORTUGAL. — Lisbon, St. f7fe, &c. : 

10 quarto = 8 meio = 4 alqueirc = 1 fanga, - = 1.534 

15 fanga = 1 majo, 4 m. = 1 last, - - = 92.087 

27 fanga = 1 tonelada,ybr shipping. 

1 haldo, for coals , - - - - - = 12. G9 

1 f\inga, ** ** - - - - = J1.1G7 

Oporto: 1 fimga = 1.937 bus. 1 raze, for salt, - = 1.25 
PRUSSIA {legal since 1820) : 4 masschen = 1 metze, 4 



FOREIGN DRY MEASURES REDUCED TO UNITED STATES, a 49 
Foreign. V, States. 

"VTinchester 
bushels. 

metze = 1 viertel, 4 v. = 1 scheffel = 3072 cubic 
Rhein zoUe or 33535- cubic inches, U. S., - - =» 1.559 

12 scheffel = 1 inalter of dromt, 2 in. ^s^^ 1 wispel, = 37.431 

3 wispel = 1 last. 

2 wispel = 1 last, for barley and oats. 
RUSSIA (legal for the Empire) : 

8 garnetz = 1 tsehetwerik, 2 t. = 1 payak, - — 1.438 
2 p. = 1 osmin, 2 o. = 1 tsehctwerk, - - = 5.952 

li tschetwerk = 1 kuhl, IG tschetwerk = 1 last. 
Lihau, Revel, Riga : 12 stof = 1 kulmct, 3 k. = 1 lof, 
24 1. == 1 tonne, 2 t. = 1 last. 
SARDINIA I. — Cagliari, &c. : 4 inibuto == 1 carbula, 

4 c. = 1 starello, 3 s. = 1 restiere or rasiera, = 4.1 GG 
SICILY I. — Pakrmo, Messina : 2 stari = 1 modello, 4 

in. = 1 tomolo, 4 t. = 1 bisaccia, 4 b. = 1 salma, = 7.81 
IG tomoli grosso = 1 salma grosso, - - = 9.72 

SPAIN. — AUcant : 2 medio = 1 celemin, 4 c. = 1 bar- 

cella, 12 b. = 1 cahiz, - ^ - - - = G.992 

Barcelona: 4 picolin = 1 cortain, 12 c. = 1 quartera, 

2i q. = 1 carga. If c. = 1 salma, - - = 8.191 

Cadiz {Standard of Casiile) : 2 medio = 1 celemin, 12 

c. == 1 fanega, 12 f. = 1 cahiz, - - - = 19.1S9 

4 cahiz = 1 last, - . - - = 7G.759 
Corunna, Ferrol : 4 celcraine = 1 ferrado, 3 f . = 1 

fanega, 12 fanega = 1 cahiz, - - - = 19.189 

Gibraltar : Same as Cadiz, 
Malaga : Same as Cadiz. 

Santander : 144 celemin = 12 fanega = 1 cahiz - = 24.984 
Valencia : 8 medio = 4 celemin = i barchilla, 12 bar- 

chille= 1 cahiz, - - - - -= 5.758 

SWEDEN. — Stockholm, &c. : 2 stop = 1 kanna, IJ k. 
= 1 kappe, 4 kappe = 1 fording, 4 f . = 1 spann, 
2 spann = 1 tunna, - - - - = 4.158 

24 tunne = 1 last. 
SWITZERLAND (legal, since 1823, for the Cantons of 
Aarau, Basle, Berne, Freiburg, Lucerne, Solotlnirn, 
Valid ; but not in general use) : 
10 emine = 1 gelt or quarteron, 10 g. = 1 sac = 

1-^jj hectolitres oiYi^n.Q'd, - - - -= 3.831 

Special and local — 
Basle: 2 bacher = 1 kopflein, 8 k. = 1 sester, - = 0.97 

4 sester = 1 sack, 2 s. = 1 vierzel, - - - = 7.75G 

Berne: 4 achterli or 2 immi = 1 massli, 2 m. = 1 

mass, 12 mass = 1 mut, - - - = 4.771 

E 



60 a FOBEIQN DBY MEASURES REDUCED TO UNITED 8TATK. 



Forc'}(jn. 

Geneva : 2 bichct = 1 coupe or sac, - - - = 

Lausanne : 10 copet = 1 emine, 10 e. = 1 sac, - - 

Neufchatel : 3 copet = I emine, 8 e. = 1 sac, - - = 

3 sacks = 1 muid. 

St. Gall : 4 massli = 1 vierling, 4 v. = 1 viertel, 4 

viertel = 1 mutt, 2 m. = 1 malter, 
Zurich : 10 massli or 4 vierling = 1 viertel, 4 v. = 1 
mutt, 4 m. = 1 malter, - - - - : 

1 mass, for salt, - _ - - : 

4 mass = 1 kurl). 

TRIPOLI (N. Africa) : 20 tiberi = 1 caffiso, - 

4 orbah = 1 tomen, 3i t. = 1 nusfiah, - = 

3 n. = 1 ui'lxa. 
TUNIS. — 12 zah or saha = 1 quiba, 16 q. = 1 caffiso 
= 18 wa^^e, - - - - - : 

TURKKY. — Constantinople: 4 kiloz = 1 fortin, - 
Latakia^ Alrppo : 1 garave, - - - -; 

Smyrna: 4 kilo = 1 fortin, - - - = 



U. Stales. 



Imshels. 

2.203 
3.831 
3.459 



= 4.G88 

= 9.333 
= 2.022 

= 1.154 
= 1.0157 



14.954 
3.7G4 

41.15 
5.824 



West Indies. 

In the Islands of AntiguGy tJie Bahamas, Barbadoes, Bar- 
Intda, Dominica, Grenada, Jamaica, Lcs Saints, 
Mantsrrrat, Nevis, St. Kitts\ St. Vincent, Tobago, 
Tortola, the Dry Measures are the same aa thoeo 
of the United States. 

In Dcscadc, Guadeloupe, Marirr/alante, Martini</ur, St. 
Lucia : 3 boisseau = 1 minut, 2 m. = 1 mine, 12 
mines = 1 setier, 2 s. ^= 1 muid, 
This ])cinn; tlio system of France before 1812. 

In Bonaire, Curacoa, Srdta, St. Eustatius, St. Martin: 
Same as in IloUand before 1820 : See IIoll.\nd. 

In Santa Cruz, St. John, St. Thomas: Same as Denmark. 

In St. Bartholomew: Same as Sweden. 

In Trinidad: Same as Castile {Spain). 

Cuba I. — Same as Castile, gcjierally. 

Havana: 4 arrobas = 1 fiincga, - - - 

Hayti I. — Aux Caycs, Cape Hayiicn, Port au Prince, 
Port Platte, &c. : Same as France before 1812. 
Savajina, St. Domingo, &c. : Same as Castile. 

Porto Rico I. — Same as Castile (Spain). 



= 53.153 



= 3.114 



:wt. 


Draft. 




lib. 


2, 


. 2 *' 


3, 


. 3 *' 


10, 


. 4 ** 


18, 


. 7 ^' 



CUSTOM HOUSE ALLOWANCES ON DUTIABLE GOODS. a 51 



CUSTOM HOUSE ALLOWANCES ON DUTIABLE GOODS. 

Draft, or Tret, is an allowance of weight for supposed waste on 
articles paying- duty by the pound. It is deducted from the actual 
gross weight of the article, and is established as follows : — 

Cv/t. 

On 1 (112 lbs.) 
Above 1 and under 
On 2 '' '' 

U ^ il it 

■*« 10 '' '' . 

** 18 '' upwards, 9 

Tare is the weight — actual or assumed by law — of the cask, 
sack, &cT, in which the article paying duty is contained. It is de- 
ducted from the actual gross weight less the draft. The remainder 
is the net weight on which the duty is assessed, and the weight at 
which the heavy purchasers receive the goods. 

Leakage is an allowance on the gauge of molasses, oils, wines, and 
all liquids in casks. It is established at 2 per cent., and is deducted 
from the actual gross gauge, less the real wants of the cask. 

Breakage is an allowance of 10 per cent, on ale, beer, and porter, in 
bottles, and 5 per cent, on all other liquors in bottles ; or, if the 
importer prefer, the duties are assessed by actual count, he so electing 
at the time of making the entry. Common sized bottles are computed 
to contain 2| gallons per dozen. 

On bottlea in which wine is imported there is assessed a duty of 
two dollars per gross, in addition to the duty on the wine. 

The following articles, whether intended for sale or otherwise, are 
admitted into the United States, from foreign ports, free of duty ; but 
nevertheless must pass through the Custom House in manner the 
same as goods on which a duty is assessed. 



Animals imported for breed. 

Antiquities. 

Bulbs or bulbous roots. 

Bullion, silver or gold. 

Canary seed. 

Cardamon seed. 

Coins, gold, silver, or copper. 

Copper sheathing, 14 by 48 inch, 

and from 14 oz. to 34 oz. per 

square foot. 
Copper ore. 
Cotton. ^ 



Cummin seed. 
Fossils. 
Gold dust. 
Guano. 

Gypsum, unground. 
Oakum and old junk. 
Oysters. 

Platina, unmanufactured. 
Silver, old, fit only for re-manu- 
facturing. 
Vanilla, plant of. 



52 a CUSTOM HOUSE allowances on dutiable goods. 



TABLE OF established TARES. 



(a=by custom; c=legal.) 


Per 
cent. 


ib«. 

per 




Per 

cent. 


ib>. 

p«r 


Almonds, in bags, 


a 4 


Pkff. 


Indigo, casks. 


cl5 


I^. 


Alum, casks. 


alO 




** zeroons, 


ClO 




Beef, jerked, drums. 




70 


Looking-glasses, Fr., 


a30 




'' '' hhds., 




112 


Mace, kegs, 


a33 




Bristles, Archanfrel, 


ali 




Nails, casks, 


c 8 




*' Cronstadt, 


al2 




Nutmegs, *' 


cl2 




Camphor, crude, tubs, 


a35 




Ochre, French, casks. 


a 10 




Candles, boxes. 


c 8 




Pepper, bales. 


c 5 




'' chests-, 160 lbs.. 


a20 




** bags. 


c 2 




Candy, su^ar, baskets, 


a 5 




** casks. 


cl2 




'' '' boxes, 


clO 




Pimento, bales, 


^ 5 




Cheese, hps. or baskets, 


alO 




baj?s. 


c 3 




** boxes, 


a20 




" casks, 


cl6 




Chocolate, boxes, 


clO 




IPruncs, boxes. 


a 8 




Cinnamon, mats, 


alO 


. 


Raisins, Malaga, boxes, 


al5 




** chests, 




16 


" casks, 


al2 




Cloves, casks, 


a\2 




jars, 


a 


18 


Cocoa, bap:s, (actual 2) 


c 1 




** Smyrna, casks, 


aV2 




'* casks, . 


ClO 




iSalls, glaubcr, casks, 


c 8 




** zeroons. 


a 8 




|Shot, casks, 


c 3 




Coffee, E. 1., p^rass hags, 




2 


Soap, French, boxes. 


al3 




** ** bales, 


c 3 




*' boxes, {a more) 


clO 




'* ** casks. 


cl2 




Steel, bundles, 


a 3 




" W.T.,bags, 


c 2 




** cases, 


a 8 




Copperas, casks. 


alO 




Sugar, bags or mats. 


c 5 




Cordage, lines, bales, 


c 3 




*' boxes. 


cl5 




Cordage, mats, 


all 




*» casks. 


cl2 




Corks, bales, light. 




5 


** canisters, 




40 


** heavy, 


(z20 




** Java, willow > 
baskets, J 




Tifl 


Cotton, bales. 


c 2 






uu 


** zeroons. 


c 6 




Tallow, casks, 


fll2 




Currants, casks, 


al2 




** zeroons, 


a 8 




J'igs, boxes, 00 lbs., 


a\0 




Tea,caddie8,^ ,( 


•*» 





'' 4 *' 30 '' 

U it 15 u 


a 6 
a3i 




o 

e3 


23 
14 


*' drums. 


alO 




Twine, bales. 


c 3 




** frails, 75 lbs., 


a i 




** casks, 


cl2 




Glue, Russia, boxes. 


al5 




Wool, Germany, bale. 


a 3 




Indigo, bags or mats, 


c 3 




** S. Amer., bale. 




15 


*' barrels. 


cl2 




** Smyrna, ** 




10 


*' cases, 


fll5 











SECTION I. 
MONEYS, WEIGHTS AND MEASURES, 

OF THE UNITED STATES ; — THEIR DEXOMIXATIOXS, VALUES, 
COMPARATIVE VALUES, MAGNITUDES, &c. 

MONEYS OF ACCOUNT OF THE UNITED STATES. 

These are the mill, the cent, the dime, and the dollar. 
10 mills s= 1 cent, 10 cents = 1 dime, 10 dimes = 1 doUar. 

The dollar is the unit or ultimate money of account of the United 
States, or of what is sometimes called Federal money. 

In practice, the dime, as a denomination of value, is rejected. 
Thus, 

10 mills = I cent, and 100 cents = 1 dollar. 

This mark, $, is equivalent to the word dollar, or dollars, in this 
money. 

COINS OF THE UNITED STATES. 

Until June, 1834, the government of the United States estimated 
gold in comparison with silver as 15 to 1, and in comparison with 
copper as 850 to 1. 

From June, 1834, until February, 1853, the same government 
estimated gold in comparison with silver as 16 to 1, and in com- 
parison with copper as 720 to 1. 

For all time since February, 1853, this government has estimated 
gold in comparison with silver as 14 J to 1, and in comparison with 
copper as 720 to 1. 

The standard for mint gold with this government until 1834, 
was 11 parts pure gold and 1 part alloy, the alloy to consist of 
silver and copper mixed, not exceeding one half copper. 

The gold coins, therefore, struck at the United States mint prior 
' to 1834, are 22 carats fine. 
2 



14 CUKRENCY OF THE UNITED STATES. 

In what, until 1834, constituted a dollar of gold coin of United 
States mintage, there were put 24.75 grains of pure gold; and 27 
grains of the standard mint gold of that day were at that time worth 
§1. Twenty-seven grains of that gold, or gold of that standard, 
are now, by the present government standard of valuation, worth 
$1.0652. 

The standard for mint silver with this government until 1834, 
was 1485 parts pure silver and 170 parts pure copper, = 8-1^7^5 
parts pure silver and 1 part pure copper. 

The silver coins, therefore, struck at the United States mint prior 
to 1834, are lOf f^^ ounces line. 

In that which, until 1834, constituted a dollar of silver coin of 
this government's mintage, there W(?re put 371.1 grains of purp 
silver; and 410 grains of the standard mint silver of that day wero 
at that time of the value of .si. Four hundred and sixteen grains 
of that silver, or silver of that standard, are noSv, by the present 
government standard of valuation, worth .<1.0744. 

The cent, until 1834, was of pure copper, and weighed 208 
grains; since 1834, pure copper, weight 1G8 grains. 

The standard for mint gold with this government is now, and for 
all time since June, 1834, has been, 9 pirts pure gold and one mrt 
alloy, the alloy to consist of silver and copper mixed, not cxceeoing 
one half silver. 

The gold coins, therefore, struck at tho United States mint and 
dated subsequent to 1834, are 21 f c;irats fine. 

The standard weight for these coins is 25 J grains to the dollar; and 
in'cvery 25| grains of these coins there are 23/(y^(y grains of pure 
gold. 

The standard for mint silver with this government is now, and 
for all time since Juno, 1834, has been, 9 parts pure silver and 1 
[)art pure copper. 

The silver coins, therefore^, struck at the United States mint and 
dated subsequent to 1834, are 10| ounces fine. 

In what, from June, 1834, until February, 1853, constituted a 
dollar of silver coin of this government's mintage, there were put 
371 1 grains of pure silver ; and 412.^ .2;rains of the standard mint 
silver of that day (the present standard) were worth, from June, 
1834, until February, 1853, $1. Four hundred twelve and one 
half grains of this standard of silver are now worth, by the present 
standard of valuation, $1.0742. 

The standard weight for silver coins with this government at 
present is 384 grains to the dollar. 

The new cent, established by the Congress of 1856, is 7 parts . 
copper and 1 part nickel, and its legal weight is 72 grains. 

The foregoing is not appUcablc to the three-cent pieces of United 



CURRENCY OP THE UNITED STATES. 



15 



States mintage. These pieces were ordered by the Congress of 1850- 
1851, and an especial standard of purity was assigned them, 
viz., three parts silver and one part copper ; their weight was fixed 
at 12f grains each, and their current value at three cents each. 
The law of 1853, regulating the currency, does not apply to these. 
They are now, as in 1851, legally the same. These pieces are worth, 
even now, less than their nominal values, compared with the present 
standard of purity and weight for other United States coins. They 
are worth, by this comparison, 2,863 cents each. 

In the preceding calculations, the alloy for gold, in each instance, 
was taken to consist of equal parts of silver and copper. The law, 
until 1834, provided that it should consist of ' silver and copper 
mixed, net exceeding one half copper;' and the present law pro- 
vides that it shall consist of ' silver and copper mixed, not exceed- 
ing one half silver, ' 

The metals used as alloys were taken at their values as money. 
Federal money was established by the Congress of tho United 
States, in 1786. 
Boston^ June^ 1866, 

GOLD, — PURE. 

24 carats fine = Pure Gold, 

1 grain = $0.0429, 

23.30859 '' = ^im. 

1 dwt. = $1.02966, 

1 ounce = $20,5932, 

MINT GOLD. — U. S. 
Alloy half each, silver and copper. 
Nine parts pure gold and one part alloy ; or, 

21f carats fine = Standard Coin, 



1 grain 


= $0 03^ 


76. 




25| ^* 


= $1.00. 




1 dwt. 


= $0.93023. 




1 ounce 


= $18.60465, 




GOLD COINS 


. — u. s. 








Weight in 


standard 






Grains. 


Value. 


Double Eagle, - - - 


• 516 


$20.00 


Eagle, 


_ 


258 


10.00 


Half Eagle, - - - . 


. 


129 


5.00 


Quarter Eagle, . - . 


. 


64^ 


2.50 


Gold Dollar, - - - - 


- 


25* 


1.00 


Triple Gold Dollar, 


- 


771 


3.00 


Eagle, prior to 1834, ($104,) - 


- 


270 


10.64 


Half do., ^' ^^ " ($5i,) - 


- 


135 


5.32 



16 



CURRENCY OF THE UNITED STATEB. 



Private and Uncurrem 


t. 


Weight 12» 
Grains. 


Sales. 


A. Bechtler, N. C, $5 piece, 




$4.75 


u li 2A *' - 


- 




2.37 


(( (( 1 *' - 


- 




.93 


T. Reed, Georgia, 5 ** - 


- 




4.75 


<< '* 2A *' - 


- 




2.37 


(( (( 1 *' - 


. 




.93 


Moffat, California, 5 " - 


- 


129 


5.00 


SILVER,— 


- PURE. 






12 ounccfl fine = 


Pure Silver. 






1 dwt. = 


$0.0G928. 






340^8 grains = 


$^1. 






1 ounce « 


§1.3857. 







MINT SILVER. — U. S. 

Alloy y all copper. 
Nino parts pure silver and one part alloy ; or, 



10 oz. IG dwts. 

1 dwt. 
384 grains 
1 ounce 



fine 



Standard Coin. 
x= .^0.0G2. 
= $1.00. 
= $1.23958. 



SILVER COINS. — U. S. 



Dollar, 

Half Dollar, . - . - . 

Quarter Dollar, 

Dime, --___- 

Half Dime, 

Three-Cent Piece, f silver and ^ copper. 



Weight In 
Grains. 



384 
192 

96 

38? 

121 



StADdard 
Value. 



$1.00 
.50 

.25 
.10 
.05 
.03 



The copper coins of the United States are the cent and half cent ; 
they are of pure copper. The weight of the former is 1G8 grains, 
and that of the latter, 84 grains. 

Note. — The silver coins of the United States, issued since February, 1863, are zMt 
legal tender iu the United States in sums exceeding^ue dollars. 



CtTRRENCY OF TUE UNITED STATES, 



17 



TABLE, 

Exhibiting the standard weight and present par value of the silver coin$ 
of the United States, of dates subsequent to 1834, and prior to 1853, 









Weight in 


Present 




. 


. 


Grains. 


par value. 


Dollar, - - . 


4124-. 


$1.0742 


HalfDoUar, - 


- 


- 


206i 


.5371 


Quarter Dollar, - 


- 


- 


103J 


.2685 


Dime, - - . 


- 


- 


41i 


.1074 


Half Dime, 


. 


- 


m 


.0537 


Three-Cent Fiece, 


- 


- 


m 


.03 



CURRENCIES OF THE DIFFERENT STATES OF THE UNION, 

4 Farthings =1 Penny, 12 Pence = 1 Shilling, 20 Shillings = 

1 Pound. 
In Massachusetts, Connecticut, Rhode Island, New Hampshire, 
Vermont, Maine, Kentucky, Indiana, Illinois, iSfissouri, Virginia, 
Tennessee, Mississippi, Texas and Florida, G shillings = 1 dollar ; 

$1 = ^3^ £. 

In New York, Ohio and Michigan, 8 shillings = 1 dollar ; $1 = 

f£. 

In New Jersey, Pennsylvania, Delaware and Maryland, 7 shil- 
lings and 6 pence = 1 dollar ; 1 dollar = |- £. 

In North Carolina, 10 shillings = 1 dollar ; $1 = J £. 

In South Carolina and Georgia, 4 shillinga and 8 pence ^^ 1 dol- 
lar; $1 = 3^^- 

Note. — These currencies^ so called, are nominal at present in a great measure. The 
denominations serve in the diflFerent States as verbal expressions of value. But they are 
neither the names of the moneys of account in any of the States, nor are they the national 
names o( any of the real moneys in circulation. All values in money in the United Statef 
are legally expressed in dollars^ cerUs^ and mills, 
2=^ 



18 METRICAL SYSTEM OF WEIGHT8 AITO MEABURW. 



THE METRICAL SYSTEM OF AVEIGHTS AND MEAS- 
URES. 

In this system, the Metre is the basis, and is one forty-millionth 
of the polar circumference of the earth. 

The Metre is the principal unit measure of length ; the Are 
o^ surface; the ^teke of solidity ; the Litre of capacity; and the 
Gram of weight. 

The gram is the weight, in a vacuum, of one cnbic centimetre 
of pure water at its maximum density. 

The Metre, almost exactly . =z 39.3085 U. S. inches. 

The Are (100 square metres) z= 3.95337 " square rods. 

The Stere (a cubic metre) . z=i 35.31042 " cubic feet. 

rvu T*. / ^r.' A ' 4. \ (C1.0164 " '' inches. 

The Litre (a cubic decmietre)iz:j ^ ^^^.^ ,, .vino quarts. 

The Gram . . . . = 15.44242 " grains. 

The divisions by 10, 100, 1,000, of each of these units, are ex- 
pressed by the same prefixes, viz., deci^ centi^ milli; and the multi- 
ples by 10, 100, 1,000, 10,000, of each, by deca, hecto, kilo, mijria. 
The former series were derived from the Latin language, the latter 
from the Greek. 

To illustrate with the metre : — 

10 771 iV/ /metres =z 1 ce;j/tmetre, 10 centimetres z= 1 (Yecfmetre, 10 
decimetres ziz 1 Metre, 10 Metres z= 1 <^/<fcametre, 10 decainc- 
tre3z= 1 7/cf/()metre, 10 hectometres i= 1 ^'//ometre, 10 kilometres 
zz: 1 77ifjriametTC. 

In commerce, the ordinary weight is the kilogram, and lOo kilo- 
grams (usually called kilos) izz 1 quintal; 10 quintals :=: 1 millicr^ 
or tonneau. 'The kilogram =: 15,442.42 -^ 7000 = 2.20C0C avoir- 
dupois pounds. 

In practice, the terms milliare, deciarcj decarc, kilnnrr. nnd myri^ 
are are usually dropped, and 

100 centare zz: 1 are ; 100 ares zz: 1 hectare. 

Also the terms milUsterc, hectostcrCj kilostcre, and myriastere^ are 
usually rejected, and 100 centisteres zz: 1 decistere ; 10 decistcres 
zz: 1 stere ; 10 steres z= 1 decastere z=z 353.1042 cubic feet. 
1 centiare (square metre) zn 1.195S9413 square yanls. 
1 kilometre . . . zi: 0.62135 statute miles. 
1 hectare . . . :=z 2.471 = U. S. acres. 

1 kilolitre . . . z= 1 stero izz G 1,010.403233 cubic in. 
A hectolitre = 26.41403 wine gallons zz 2.83741 Winchester bush. 

Note. — The system is the one recommended by the Statistical Conjarress of 
1S65 as a general system of weights and measures to be adopted by all iLatlou&. 



BOREIGN GOLD COINS. 



19 



FOEEIGN GOLD COINS. 



Note. — The coins of any country, both gold and silver, circulating as for- 
eign in any other, particuhirly those of the smaller denominations, are usually 
held at an estimate below their standard par value, compared with the money 
^andard of the country in which they circulate as foreign. Many of them, 
more particularly the silver, having circulation in the United States, are much 
worn and otherwise depreciated. In some instances, owing to frequent changes 
made both with regard to weight and purity, certain of them, having the same 
name and general appearance, bear a premium at home; others, a discount. 
Others, again, can hardly be said to have a definable value anywhere. The par 
value of the old pistole of Geneva, for instance, weighing 103i grains, is $3,985, 
while tliat of the new, weighing 87ii grains, would, at the same degree of 
purity, be worth but $3.38G ; whereas, owing to its higher standard of fine- 
ness, its par value is $3,443. The ducat of Austria, coined in 1831, weighs 
")3i grains, — its purity is 23.(14, and its par value $2.2(i'J; while the half 
sovereign, closely resembling the ducat, coined in 1835, and weighing 87 grains^ 
has a purity only of 21. ()4, and a par value, consequently, of but $3,378. The 
circulating value of the ducat in the United States, in general, is $2.20, and 
that of the half sovereign of Austria, $3.25. 



ARGENTINE REPUBLIC. 

Doubloon to 1832, 

" to '' 


Standard 

of 
purity in 


Standard 
weight 

in 
•Trains. 


Par value 

in 

Federal 

money. 


Circulating 
value in 
Federal 
money. 


Par va^- 

ue per 
g^ram. 
cts. 


19.56 
20.83 


418 
415 


$14,671 
15.512 


$ 


3.50 
3.73 


AUSTRIA. 












Sovereign y half in propor- 
tion, to 1785, 

Sovereign, half in propor- 
tion, since 1785, 

Ducat, double in propor- 
tion, 


22.00 
21.64 
23.64 


170 
174 
53i 


6.711 
6.756 
2.269 


6.50 
6.50 
2.20 


3.94 
3.88 
4.24 


BELGIUM. 












Sovereign, half in pro.. 


22.00 


170 


6.711 




3.94 



20 



FOKEIGN GOLD COIN? 





Staftdard 


Standard 


Par Talue 


Circulating 


Par T»l- 




of 


weight 


in 


ralue in 


ue per 




purity in 


in 


Federal 


Federal 


<-rain. 


Twenty Franc, inore in pro. 


carats. 


jrrainB. 


money. 


money. 


Clt. 


21.50 


994 


$3,840 


$3.83 


3.85 


Ducat, 








2.20 




Bolivia, Colombia, Chili, 












Ecuador, Peru, New 












Grenada, and Mexico, 












Received by U. S. Gov- 












ernment, — tlioae of not 












less than 20.86 carats 












fine, at SO-j-^^ cts. per dwt. 












Doubloon, (8 E) 


20.86 


417 


15.620 


15.60 


3.74 


Half do. 


11 


208i 


7.810 


7.50 


t( 


Quarter do. 


n 


104 i 


3.905 


3.75 


t( 


Eighth do. 


n 


52 


1.952 


1.75 


(( 


Sixteenth do. 


a 


26 


.976 


.90 


(( 


Pistole, half in pro., 








3.75 




BRAZIL. 












Received by the U. S. 












Government, — those of 












not less than 22 carats 












fine, at 94^"q cls. per dwt. 












D()braon, 


22.00 


828 


32.711) 


32.00 


3.95 


Dobra, 


(t 


438 


17.306 


17.00 


i* 


Joannes, {standard variable) 


(( 


432 


17.004 


8l3totl7 


ti 


Half do. do. do. 


(( 


210 


8.532 


96 to 8.50 


»< 


Moidore, (BBBB) half in 












pro., (standard variahk) 


21.79 


165 


6.451 


6.00 


3.90 


Crusado, do. do. 


(( 


m 


.636 




(( 


DENMiVK. 












Christian d'or 


21.71 


103 


4.01R 




3.90 


Ducat, species. 


23.48 


534 


2.254 


2.20 


4.21 


** current, 


21.03 


48 


1.811 




3.77 


FRANCE. 












{Alloy fnostli/ silver,) 












Rcc'd by U. S. Govern- 












ment, — those the purity 












of which is not less than 












21y^^ carats fine, at 93y\j 












cts. per dwt. 












Chr. d'or, double in pro., 


21.60 


101 


3.014 


3.90 


3.87 



FOREIGN GOLD COINS. 



"M 





Standard 


Blaridard 


Par value 


Circulating 


Par Tal- 

U8 per 




of 


weight 


in 


value in 




purity in 


in 


Federal 


Federal 


grain. 


Franc d'or, double in pro., 


carata. 


grains. 


money. 


money. 


ctt. 


21.60 


101 


$3,914 


$3.90 


3.87 


Louis d'or, " '' " 












to 1786, 


21.49 


125i 


4.840 




3.85 


Louis d'or, double in pro., 












since 1786, 


21.68 


118 


4.573 


4.50 


3.87 


Napol©3n(20F.) double &c. 


21,00 


99i 


3.856 


3.83 


(( 


GERMANY. 












BADEN. 












Zehn Gulden, 5 in pro.. 


21.60 


105i 


4.088 


4.00 


3.87 


BAVARIA. 












Carolin, 


18.49 


149J 


4.952 




3.32 


Ducat, double in pro., 


23.58 


53| 


2.275 


2.20 


4.23 


Maximilian, 


18.49 


100 


3.317 




3.31 


BRUNSWICK. 












Ducat, 


23.22 


53.i 


2.220 




4.16 


Pistole, double in pro.. 


21.00 


1171 


4.548 




3.87 


Ten Thaler, 5 in pro., to 












1813, 


21.55 


202 


7.811 


7.80 


3.86 


Ten Thaler, less in pro.. 












since 1813, 


21.50 


204 


7.873 


7.80 


3.85 


HANOVER. 












Ducat, double in pro.. 


23.83 


53.i 


2.287 


2.20 


4.27 


George d'or, *' " ** 


21.G7 


102i 


3.987 




3.88 


Zehn Thaler, 5 '' '' 


21.36 


204i 


7.838 


7,80 


3.83 


HESSE, 












Ten Thaler, 5 in pro., to 












1785, 


21.36 


202 


7.742 




{< 


Ten Thaler, 5 in pro. , since 












1785, 


21.41 


203 


7.799 




3.84 


SAXONY. 












Ducat, 


23.49 


53i 


2.256 


2.20 


4.21 


Augustus d'or, double in 












pro., since 1784. 


21.55 


102i 


3.964 




3.86 


WURTEMBURG. 












Carolin, 


18.51 


1474 


4.899 




3.32 


Ducat, 


23.28 


53i 


2.235 




4.17 



22 



FOEEIGN GOLD COINS. 





Scindard 


Standard 


Par Talue 


Circulating 


Par val- 




of 


wcifht 


in 


▼ alue in 


ue per 




purity in 


in 


Feder^ 


Federal 


^ain. 


GREAT BRITAIN. 


c.irats. 


graini. 


money. 


money. 


cts. 












(Alloy, since 1826, all copper.) 












Rec'd by U. S. Govern- 












ment, — those of 22 ca- 












rats fine, at 9i^jj cts. 












per dwt. 












Guinea, half in pro., to 












1785, 


22.00 


127 


$5,016 




3.95 


Guinea, half in pro., since 












■ 1785, 


{( 


129i 


5.111 


$5.00 


ii 


Sovereign, half in pro., 


(( 


123i 


4.800 


4.83 


(( 


Five do. 


i( 


610.1 


21.332 


24.20 


{( 


Sovereign, (dragon) half 












in pro., 


(( 


122i 


4.838 


4.80 


(( 


Double Sovereign (dragon) 


(( 


216 


9.717 


9.67 


(( 


GREECE. 












Twenty Drachm, more in 












pro.. 


21.00 


69 


3.449 


3.30 


3.87 


HOLLAND. 












D'lcat, 


23.58 


534 


2.203 


2.20 


4.«3 


Ryder, 


22.00 


153 


6.043 




3.95 


Double do. 


(i 


309 


12.205 




*i 


Ten Gulden, 5 in pro., 


21.60 


101 


4.025 


4.00 


3.87 


INDIA. 












Pagoda, star, 


19.00 


521 


1.798 




3.40 


Mohur, (E. I. Co.) 1835. 


22.00 


180 


7.100 


6.75 


3.95 


Half Sovereign, do. 








2.41 




BOMBAY. 












Rupee, 


22.09 


179 


7.095 




3.96 


MADRAS. 












Rupee, 


22.00 


180 


7.100 




3.95 


ITALY. 












Eturia, Ruspone, 


23.97 


1011 


6.935 




4.30 


Genoa, Sequin, 


23.86 


53.^ 


2.291 




4.28 


Milan, Pistole, 


21.76 


97i^ 


3.807 




3.90 


** Sequin, 


23.70 


53i 


2.281 




4.26 



FOREIGN GOLD COINS. 



23 



Milan, Twenty Lire, more 

in proportion, 
Naples, Ducat, multiples 

in pro., 
Naples, Oncetta, 
Parma, Doppia, to 1786, 
'' Pistole, since 1796, 

** Twenty Lire, 
Piedmont, Carlino, half in 

pro., since 1785, 
Piedmont, Pistole, half in 

pro., since 1785, 
Piedmont, Sequin, half in 

pro., since 1785, 
Piedmont, Twenty Lire, 

more in pro., 
Rome, Ten Scudi, 5 in pro. 

" Sequin, since 1760, 
Sardinia, Carlino, i in 

pro., 
Tuscany, Zechino, double 

in pro., 
Venice, Zechino, double 

in pro., 

MALTA. 
Sequin, 

Louis d'or, double and 
demi in pro., 

NETHERLANDS. 
Ducat, 
Zehn Gulden, 5 in pro.. 



Ducat, 

PORTUGAL. 

Rec'd by the U. S. Gov- 
ernment, — those t lie pu- 
rity of which is not less 
than 22 carats fine, at 
94y\5- cts. per dwt. 



Sundard 

of 
purity ill 

carats. 



StAndard 
•rti^hl 



21.58 

21.43 

23.88 
21.24 
20.95 
21.62 

21.69 

21.54 

23.64 

20.00 
21.60 
23.90 

21.31 

23.80 
23.84 



Par Taluc 

in 

Federal 



23.52 
21.55 



23.58 



99i 

22i 
58 

110 

110 
994 

702 

140 

53i 

99i 

2674 

524 

2474 

53? 

54 



23.70 534 
20.25 128 



534 
103 



534 



$3,853 

.865 
2.485 
4.192 
4.135 
3.860 

27.321 

5.411 

2.280 

3.503 

10.368 

2.251 

9.465 

2.302 

2.310 

2.275 
4.651 



2.257 
4.013 



2.264 



Circulating 
wlue in 
Federal 
money. 



$3.83 



3.83 



3.50 



4.00 



Par Tal- 
ue per 



3.86 

3.84 
4.28 
3.81 
3.75 
3.87 

3.89 

3.86 

4.23 

3.59 

3.87 
4.28 

3.82 

4.30 



4.25 

3.63 



4.21 

3.86 



4.23 



24 



rOREIGN GOLD COIWS. 



■■' 


Standard 


Standard 


1 Par Ttilu* 


Circtilatjng^ 


F-TTrf. 




of 


weight 


m 


Tilne in 


v*ptm 




purity in 


in 


1 Federal 


Federal 


irraiB. 


Dobraon, 24,000 reis, 


cardta. 


l?T«ini. 

828 


moner. 


monej. 


cf. 


22.00 


$32,706 


$39.00 


3.95 


Dobra, 


(( 


438 


17.301 


17.00 


(( 


Joannes, {standard variable) 


(( 


432 


17.064 


•13toS17 


i« 


Half '* 


a 


216 


8.532 


#6 to 6.50 


(( 


Moidore, 4000 reis, ** 








•41 to 941 




Coroa, 5000 " 


C( 


147i 


5.R3 


5.75 


(( 


Milrea, 


22.00 


19i 


.780 




3.95 


FRUSSIA. 












Ducat, 


23.49 


534 


2.255 


3.20 


4.21 


Frederick d'or, double in 












pro., 


21.60 


102i 


3.973 




3.87 


RUSSIA. 












Ducat, 


23.64 


!>l 


2.291 




4.2^1 


Imperial, (10 R.) half in 












pro., 1801, 


23.r>5 


185.i 


7.828 




4.22 


Imperial, (10 R.) half in 












pro., since 1818, 


22.00 


199 


7.856 


7.80 


3.95 


SICILY. 












Oncia, double in pro.. 


20.39 


68i 


2.495 




3.64 


Twenty Lire, more in pro.. 


21.60 


994 


3.856 


3.83 


3.87 


SPAIN. 












RecM by U. S. Goven>- 












mont, — those the stand- 












ard purity of which is 












not less than 20.86 ca- 












rats fine, at 89y^^ cts. 












per dwt. 












Doubloon (8 S) parts in pro. 


21.45 


4164 


16.031 


16.00 


3.84 


'* (8 E) parts as 












Bolivian, &c. 


20.86 


417 


15.620 


15.60 


3.74 


Pistole, to 1782, 


21.48 


103 


3.970 




3.85 


** since *' 


20.93 


104 


3.906 




3.75 


Escudo, to 1788, 


20.98 


52 


1.957 




3.76 


" since '' 


20.42 


52 


1.905 




3.66 


Coronilla " 1800, 


20.29 


27 


.983 




3.64 


SWEDEN. 












Ducat, 


23.45 


53 


2.230 




4.20 



LONG Oa Lm£A£ MEASURE. 



25 



SWITZERLAND. 
Berne, Ducat, double in 

pro., 
Berne, Pistole, 
Geneva, Pistole, 

'' *' {old) 

Zurich, Ducat, double in 
pro., 

TURKEY. 
Misseir, half in pro. 1820, 
Sequin foiiducli, 
Yeermecblekblek, 



Standard 


Sundard 


of 


weight 


puriiy in 


in 


carali. 


irraini. 


23.53 


47 


21.62 


1174 


21.87 


87j 


21.51 


1031 


23.50 


53.i 


15.88 


3H.i 


19.25 


53 


22.88 


733 



Par value 


Circulating 


in 


Talue in 


Federal 


Federal 


monfT. 


money. 


$1,984 




4.558 




3.443 




3.985 




2.25G 




1.040 




1.830 




3.027 





PaxTal 
le par 

pram. 



4.22 

3.88 
3.92 
3.85 

4.21 



2.84 
3.45 
4.10 



Note. — The Milled Dollars, ox Pesos (silver) of Spain, Mexico, Peru, Chili, and Cen 
tral America, wnd the restanifMxl of BrazU, weighing not leas than 415 j^rains, and of 10 
oz. 15 dwt.s. fine, are received by the Unite<l Slates Government at ^\Si) eacli. 

The Five Franc silver pieces of France, of 10 oz. 16 dwl«. fine, and weighing 384 
grains, are received at 9"i cents each. 

The standard silver coins of Great Britain are 11 oz. 2 dwts. fine. 



LONG OR LINEAR MEASURE. — U. S. 

Standard. — A brass rod, the length of which, at 62^ Fahrenheit, 

is §§:yJo J t.hat of a pendulum beating seconds in vacuo^ at the level 
ofihe sea, at the latitude of London, = 5|;^*^^ J at 32'' Fall., at the 
gravitation at New York, = the Yard. 



6 points 

12 lines (72 points) 
12 inches 

3 feet (36 inches) 



2.1 inches 

4 nails (9 inches) 



= 1 line. 
= 1 inch. 
= 1 foot. 
= 1 yard. 



5.i yards (16.ift.) = l rod. 
40 rods (220 yds.) == 1 furlong. 
8 fur. (5280 feet) = 1 stat. mile. 



SPECIAL, FOR CLOTH. 

: 1 nail. I 4 quarters (36 inches) 

1 quarter. | 



: 1 yard. 



"^Tinj inches 
25 links 



= 1 link. I 100 links (66 feet) = 1 chain. 
= 1 rod. I 80 chains (320 rods) = 1 s. mile. 

engineer's chain. 

10 inches = 1 link. 

120 links (100 feet) = 1 chain. 
3 



f 



2S SQUARB OB SUPERFICIAL MEASURE. 

SHOEMAKER'S MEASURE. 
No. 1 is 4 J inches in length, and each succeeding number is an 
addition of J of an inch. No. 1 man's size = 844 inches. 

MISCELLANEOUS. 

Fathom = 6 feet. 



Hair's breadth —^V i^^^- 

Digit = 10 lines. 

Palm = 3 inches. 

Hand = 4 ** 

Span = '' 



Knot == 47J feet. 

Cable's length = 120 fathom*. 
Geometrical pace =4.4 feet. 



12 particular things = 1 dozen. 
12 dozen (144) =1 gross. 

12 gross (1728) = 1 great gross. 
20 particular things = 1 score. 
24 sheets of paper = 1 quire. 
20 quires = 1 ream. 

SQUARE OR SUPERFICIAL MEASURE. 
{Length X breadth.) 
144 square inches = 1 square foot. 
9 ** feet =1 *» yard. 
30i '* yards =1 ** rod. 
40 ** rods == 1 rood. 

4 roods = 1 acre. 

SPECIAL, FOR LAND. 

62||^^ square inches = 1 square link. 
10000 *' links = 1 '' chain. 

10 **• chains = 1 acre. 

Square rod = 272 i square feet. 

Rood 



1210 "• yards. 
10890 ** feet. 



Acre (160 square rods) == ^ [^^^^^ «« fe^t. ' 

cj -1 S 640 acres. 

Square mile = \iO%m sq. rods. 

220 X 198 square feet "] 

The square of 12.649 *» rods I . 

" ** of 69.5701 *' yards ?" — ^ acre. 

" " of 208.710321 ** feet J 



CUBIC OR SOLID HIEASURE. 



27 



CIRCULAR MEASURE. 



Minute, or 
Geo^a- 
phicalm. 

(60^0 
League 

Degree 




1.152 8. miles. 
6086 feet. 



: 3 miles. 

560 geo. miles. 
69.158 8. ms. 



24897 8. m. 



Great Circle = 360 degrees. 
Equatorial cir- 
cumference 
of the earth 
Equatorial diam.= 7925 
Polar diam. = 7890 

Mean radius = 3955.92 



H 



Sign(y^ zod.)=30 degrees. 

Note. — In the expressions, square feel and feet square, there is this difference; viz., 
the former expresses an area in which there are am many square feet as the number 
named, and the latter an area in which there arc as many square feet as the square of tha 
number named. The former particularizes no form of area, the latter asserts a squart 
form. 

CUBIC OR SOLID MEASURE. — U. S. 



{Length X breadth X depth.) 

; 1.273 cylindrical feet. 
22(T0 *' inches. 

3300 spherical '* 

6600 conical '' 

{0.785398 cubic feet. 
1357.2 '' inches. 
2592 spherical '' 
5184 conical " 

1 cubic yard, 
of round timber = 1 ton. 






Cubic foot, 
1728 cu. inches 



CyTmdrical foot 
1728 «' inches 

27 cubic feet 
40 

42 " of shipping * 

50 '< of hewn 

128 '' 

Cubic foot of pure water, 
at the maximum density 
at the level of the sea, 
(39^.83, barometer 30 
inches) 

Cylindrical foot 

Cubic inch 



= 1 ton. 
= 1 ton. 
= 1 cord. 



__ < 62J avoirdupois pounds. 



^ 1000 



ounces. 



Cylindrical inch 

Pound 

*' distilled = 

Cubic inch " == 

Pound at 62°, distilled = 

Cubic inch at 62°, " = 

" " 39^.83, in vacuo = 



( 0.03 

= { 0.57 

(253. 



49.1 *« 

85.4 *< 

036169^' 
87 '' 
1829 

028415 avd. 

4546 '* 
27.648 cubic inches. 
27.7015 <' " 
252.6839 grains. 
27.7274 cub. inches. 
252.458 grains. 
253.0864 *« 



pounds. 

ounces. 

pounds. 

ounces. 

grains. 

pounds, 

ounces. 



Cubic foot of salt water (sea) weighs 64.3 pcmnds. 



28 



GENERAL MEASITRE OP WEIGHT. 



GENERAL MEASURE OF WEIGHT. — U. S. 



AVOIRDUPOIS. 

Standard. — The pound is the 
weight, taken in air, of 27.7015 
cubic inches of distilled water at 
its maximum density, (39^.83 F., 
the barometer being at 30 inches) 
= 27.7274 cubic inches of distilled 
water at 62^ = 7000 Troy grains. 

27^2" grains = I dram. 

IG drams ( 137i grs.) = 1 ounce. 
10 ounces (7000 grs.)= 1 pound. 



SPECIAL 



GROSS. 



SPECIAL TEGY. 

(Exclusively for gold and sil- 
ver bullion, precious stones, and 
gold, silver and copper coins, and 
with reference to their monetary 
value only.) 

24 grains «» 1 pennyw't. 

20dwts. (480 gr8.)«= 1 ounce. 
12 oz. (5760 grs.) ^ 1 pound. 

SPPXIAL APOTHECAKIES'. 

(Exclusively for compounding 
medicines, for recipes and pre- 
scriptions.) 
20 grains = 1 scruple, 9. 

3 scruples = 1 dram, 5. 

8 drams(180 g.)= 1 ounce, S. 
12 oz. (5760 g.) = 1 pound, lb. 

1 lb. avoir. = 1-jJ^ lbs. troy. 

1 lb. troy = i^^ lbs. avoir. 

1 oz. avoir. = \^^ oz. troy. 

1 oz. troy = l^y^ oz. avoir. 

• lily is as the siiuarB of 
irai in the rough sUla, 

ICO Uial weight to makii 
one when workoil down (Hiual lo 1 carat in welghU Hence, to determine the value of a 
wn>ucht diamond of any givpn numlwr of carata : — Ruic — Double the weight in caraU 
and niuliiply the tK|uarc by 9.50. Thus, the value of a wrought dlaiaood. welirhiiur 2 
caralij, i3 2-f 2=lX4 = 16X9.5<'=ila2. i '-w -• 



28 pounds = 1 qnnrtar. 

4 quarters ) ^ I (luinlal. 

112 pounds \ ^1 cwt. 

20 cwt. = 1 ton. 

SPECIAL DIAMOND. 

16 parts = 1 grain = 0.8 troy gr. 
4 grs. = 1 carat = 3.2 *' " 



Note, — The commfalivo \.uur .m 

their respective weights. A diamond « 

is c-slinwiled worth alwut ^-N'^g^ ; not 



LIQUID MEASURE.— U. S. 

The ** Wine" or '' Winchester'' Gallon, of 231 cubic inches 
capacity, is the Government or Customs gallon of the United States 
for all liquids, and the legal gallon of each state in which no law 
exists fixing a state or statute gallon of its o\yn. It contains 58373^ 
grains of distilled water at 39^.83, the barometer being at 30 inches. 

4 gills = 1 pint, 2 pints = 1 quart. 
4 quarts, or 231 cubic in. ^ — 5 ^ R^^^lon. 

1. ( ^ 8.355 i 



0.13308 cub. ft., 294.1176 cyl. in. 



avM. lbs. pure water. 



Liquid gallon of the 
State of New York 
281.62 cylindric in 



DRY MEASTTKE. 29 

0.128 cubic foot. 

rater 



'^♦1=J 221.184*' in. 

^' ( J 8 avoid, lbs. pure wa 

• ^ Lat39°.83, b. 30 in. 



Barrel = 31J gallons. I Puncheon = 84 gallons. 
Tierce = 42 *' Pipe or Butt = 126 " 

Hogshead = 63 ** | Tun = 252 " 

Imperial gallon, > O^ ^^'^ ^^^- distilled water 

277.274 cub. in. J ^ at 62^ F., b. 30 in. 

Ale gallon, ^ _. J 10^ av'd lbs. pure water 
282 cub. in. J ^ at 39^.83, b. 30 in. 



(o.e 

on = { 0.8 

(o.i 



0.8331 Imperial gallon. 
1 Wine gallon = { 0.8191 Ale " 

.10742 W. bushel. 



1 Imperial gallon = 1.2 Wine gallons. 



DRY MEASURE. — U. S. 

The " Winchester Bushel,'' so called, of 2150^*2^ cubic inches 
capacity, is the Government bushel of the United States, and the legal 
bushel of each state having no special or statute bushel of its own. 
The standard Winchester bushel measure is a cylindrical vessel hav- 
ing an outside diameter of 194 inches, an inside diameter of ISJ 
inches, and an inside depth of 8 inches. The standard " heaped " or 
*' coal" bushel of England was this measure heaped to a truo cone 6 
inches high, the base being 19i inches, or equal to the outside diam- 
eter of the measure. Its ratio to the even bushel was, therefore, as 
1.28, nearly, to 1. The present " Imperial " measure of England has 
the same outside diameter and the same depth as the Winchester, and 
an internal diameter of 18.8 inches, and the same height of cone is 
retained for forming the heaped bushel. Its ratio, therefore, to the 
even bushel is a trifle less than was that of the Winchester. In the 
United States the *' heaped bushel " is usually estimated at 5 even 
pecks, or as 1.25 to 1 of the standard even bushel, which, if taken as 

* By enactment of the Legislature of the State of New York, this gallon ceased to be the 
legal gallon of that State, April 11, 1852 ; and the United States Goyemment gallon, of 231 
cubic inches capacity, was adopted in its stead. 

3* 



30 



DRY JIEASURE. 



the rule, requires a cone on the Winchester measure of 5.4 inches to 

equal the heaped AVinchester bushel. 



4 gills . 
2 pints . 

4 quarts 

8 quarts 

4 pecks 1 

2150.42 cubic in. ( 

1.244456 '' ft. f 

1.5814 ryl. *' J 

Bushel of the ) 

State of New York*} 
2810.1955 cyl. in. ) 

Bushel of Connecticut 

Heaped Win. bushel 
1.28— even" ** 
Imperial bushel 
Chaldron 

1 Winchester bushel 

1 Imperial bushel 



= 1 pint. 
= 1 quart. 

__ {I gallon 

^ or half peck. 
= 1 peck. 

{1 bushel. 
2738 cyl. in. 
77.7785 av'd lbs. 
pure water. 

1.28 cubic feet, 
in. 



( 1.28 cubic f 
= { 2211.84 **i 
( 80 avM lbs. 
( 1.272 cul 

,t= <2iys - 

( 79.50 av' 



pure water, 
cubic feet. 



m. 
M lbs. pure water. 
J 2747.7 cubic in. 
^ 1.59 cubic ft. 
= 2218.192 *Mn. 
= 3(J Winch, heaped bushela. 
__ K 0.9004 Imperial bushel. 
— } 9.3092 Wine pillonfl. 
= 1.0315 Winchester bushels. 



Note. — The Imperial bushel, montioned abovo. i^ tho nr.^xfnt l.L'nl hushd of Great 
Britiiin ; and the Imperial gallon, nionUonc<l on • Ije preaeot legal 

gallon of Great Britain, for all li'inids. The gul :ne as the gallon 

for dry measure. Ki^'ht Im|H."rial jrallons make ou- .-u^ii- 1. —? nf t^ gal- 

lon and the bushel, and their denonjinations, arc the same as 
In Great Britain, iu addition to the denominatioos of dry r 
Btotcs, the 



io United 



Strike, = 2 bushels. 

Coomb, = 4 " 

Quarter, = 8 " 

Wey or load, = 40 " 



Last, 

Sack of corn, . . . 
IV)Ie of corn, . . . 
Last of guniwwder, 



. = 80 hoabeli. 

.= 3 " 
.= 6 " 
. = 42 barrcU. 



* This buahel ceased to be the legal bushel of this State April 11, 1852, and the United 
States Government bushel, of 2ldO-^j^jj cubic iDches capacity, was adopted aa the kgid 
bushel In its stead. 

t This bushel is now, January, 1S52, no longer the legal buahel of this State, and the 
standard Winchester bushel is adopted in its stead. 



SECTION II. 

MISCELLANEOUS FACTS, CALCULATIONS, AND PRACTICAL 
MATHEMATICAL DATA. 



SPECIFIC GRAVITIES. 

The specific gravity of a body is its weight relative to the weight 
of an equal bulk of pure water at the maximum density, (39°. 83, b. 
30 in.) the water being taken as 1., a cubic foot of which weighs 
1000 avoirdupois ounces, or 62^ lbs. The specific gravity, therefore, 
of any body multipjied by 1000, or, which is the same thing, the dec- 
imal being carried to three places of figures, or thousands, as in the 
following TABLES, the whole taken as an integer equals the number 
of ounces in a cubic foot of the material : multiplied by G2.5, or con- 
sidered an integer and divided by 16, it equals the number of pounds 
in a cubic foot; and multiplied by .036169, or taken as an integer 
and divided by 27648, it equals the decimal fraction of a pound per 
cubic inch ; by which, it is readily seen, the specific gravity of a 
commodity being known, its weight per any given bulk is easily and 
accurately ascertained ; as, also, its specific gravity, the weight and 
bulk being known. The weight of any one article relative to that of 
any other, is as its respective specific gravity to the specific gravity 
of the other. 



METALS. 


Specific 
giaviiy. 




Specific 
gravity. 


Antimony, . 


6.712 


Gold, pure, hammered 


19.546 


Arsenic, 


5.810 


Iridium, 


15.363 


Bismuth, 


9.823 


Iron, cast, 


7.209 


Bronze, 


8.700 


*' wrought, . 


7.787 


Brass, best, . 


8.504 


Lead, 


11.352 


Copper, cast. 


8.788 


Mercury, 32^, . 


13.598 


" wire-drawn, 


8.878 


60^ . 


13.580 


Cadmium, . 


8.60-4 


u _39o^ . 


15.000 


Cobalt, 


7.700 


Manganese, 


8.013 


Chromium, 


5.900 


Molybdenum, 


8.611 


Glucinium, 


3.000 


Nickel, 


8.280 


Gold, pure, cast, . 


19.258 


Osmium, . 


10.000 



32 



SPECIFIC GRAVITIES. 





Speeific 






ffravity. 




Platinum, cast, . 


19.500 


Granite, red. 


** hammered, 


20.337 


*' Lockport, 


*' rolled. 


22.069 


'* Quincy, 


Potassium, 60^, . 


0.865 


'* Susquehanna, 


Palladium, 


11.870 


Grindstone, . 


Rhodium, 


11.000 


Gypsum, opaque. 


Silver, pure, cast, 


10.474 


Hone, white. 


** hammered, 


10.511 


Hornblende, 


Sodium, . 


0.970 


Ivory, 


Steel, soft. 


7.836 


Jasper, 


*' tempered. 


7.818 


Limestone, green. 


Tin, cast, 


7.291 


'' white, 


Tellurium, 


6.115 


Lime, compact, . 


Tungsten, 


17.600 


'' foliated, . 


Titanium, 


4.200 


'' quick. 


Uranium, 


9.000 


Loadstone, . 


Zinc, cast, 


6.861 


Magnesia, hyd., . 
Marble, common, 


STONES AND EA 


RTHa. 


** white Ital. 


Alabaster, white, 


2.730 


** Rutland, Vt., . 


*' yellow. 


2.699 


'' Parian, . 


Amber, 


1.078 


Nitre, crude. 


Asbestos, starry. 


3.073 


Pearl, oriental, 


Borax, 


1.714 


Peat, hard, 


Bone, ox, . 


1.656 


Porcelain, China, 


Brick, 


1.900 


Porphyra, red, . 


Chalk, white, . 


2.782 


** green. 


Charcoal, . 


.441 


Quartz, 


" triturated. 


1.380 


Rock CrysUil, 


Cinnabar, . 


7.786 


Ruby, 


Clay, . . 


1.934 


Stone, common, . 


Coal, bitum. avg., 


1.270 


*' paving. 


'' anth. ** 


1.520 


** pumice. 


Coral, red. 


2.700 


'* rotten. 


Earth, loose. 


1.500 


Salt, common, solid, . 


Emery, 


4.000 


Saltpetre, refined, 


Feldspar, 


2.500 


Sand, dry, . 


Flint, white. 


2.594 


Serpentine, 


" black. 


2.582 


Shale, 


Garnet, 


4.085 


Slate, 


Glass, flint, 


2.933 


Spar, fluor. 


*' white. 


2.892 


Stalactite, . 


" plate, 


2.710 


Tale, black, 


" green, 


2.642 


Topaz, 



SPECIFIC OEAVITIES. 



33 





Specific 




specific 




gravity. 




gravity. 


SIMPLE SUBSTANCES, 


Pine, yellow. 


.568 


neither metallic nor ^ 


gaseous. 


Poplar, white. 


.383 


Boron, 


1.968 


Plum, . 


.785 


Biomine. 


2.970 


Quince, 


.705 


Carbon, 


3.521 


Spruce, white, 


.551 


Iodine, 


4.943 


Sassafras, 


.482 


Phosphorus, 


1.770 


Sycamore, 


.004 


Selenium, . 


4.320 


Walnut, 


.671 


Silicon, 


1.184 


Willow, 


.585 


Sulphur, 


1.990 


Yew, Spanish, 


.807 






'' Dutch, 


.788 


WOODS, {dry. 


) 






Apple, 


0.793 


Highly seasoned Am, 


Alder, 


. • .800 


Ash, white, . 


.722 


Ash, . 


.700 


Beech, 


.624 


Beech, 


.696 


Birch, 


.526 


Birch, 


.720 


Cedar, . 


.452 


Box, French, 


1.328 


Cherry, 


.606 


'' Dutch, 


.912 


Cypress, 


.441 


Cedar, 


.561 


Ehn, . 


.600 


Cherry, 


.715 


Fir, 


.491 


Chestnut, 


.610 


Hickory, red, 


.838 


Cocoa, 


1.040 


Maple, hard. 


.560 


Cork, 


.240 


Oak, white, upland. 


.687 


Cypress, 


.644 


*' James River, 


.759 


Ebony, American, 


1.331 


Pine, yellow. 


.541 


" foreign, . 


1.290 


** pitch, . 


.536 


Elm, . 


.671 


*' white, 


.473 


Fir, yellow. 


.657 


Poplar, (tulip,) 


.587 


" white. 


.569 


Spruce, white. 


.465 


Hacmetac, . 


.592 






Hickory, red. 


.900 


GUMS, FATS, &C 




Lignum vitae. 
Larch, 


1.333 
.544 


Asphaltum, . 


5 .905 
\ 1.650 


Logwood, . 


.913 


Beeswax, 


.965 


Mahogany, Spanish, be 


5t, 1.065 


Butter, 


.942 


*' " COl 


ti., .800 


Camphor, 


.988 


*' St. Doming 


0, .720 


Gamboge, 


1.222 


Maple, red. 


.750 


Gunpowder, . 


.900 


Mulberry, . 


.897 


" shaken. 


1.000 


Oak, live, . 


1.120 


'* solid. 


5 1.550 
\ 1.800 


'' white. 


.785 


Orange, 


.705 


Gum, Arabic, 


1.454 


Pear, 


.661 


^* Caoutchouc,. 


.933 


Pine, white, 


.554 


'^ Mastic, 


1.074 



34 



SPEClflC GKAVITIES. 



Honey, 
Ice, 

Indigo, 

Lard, 

Pitch, 

Rosin, 

Spermaceti, 

Starch, 

Sugar, dry. 

Tallow, 

Tar, . 

LIQUIDS. 

Acid, acetic, 
*' citric, 
'* fluoric, 
** nitric, 
** nitrous, 
'* sulphuric, 
** muriatic, 
'* silicic, 
Alcohol, anhyd. 

** 90 7o 
Beer, 

Blood, human, 
Canipheno, pure. 
Cider, whole. 
Ether, sulph., 
'* nitric. 
Milk, cow's, 
Molasses, 75 ''/o 
Oils, linseed, 
** olive, 
*' rapeseed, 
** sassafras, 
^* turpentine, com. 
** sperm, pure, 
** whale, pT'd, 
Proof spirits. 
Vinegar, 
Water, pure, 
*' sea, 
** Dead sea, 



specific 

graviiy. 

1.450 

.930 
1.009 

.911 
1.150 
1.100 

.913 
1.530 
1.G06 

.938 
1.015 



1 .002 

1.031 

1.000 

1.1H5 

1.120 

l.HW) 

1.200 

2.r.(i0 

.794 

.834 

1.031 

1.054 

.803 

1.018 

.715 

.908 

1.032 

1.400 

.934 

.917 

.927 

1.090 

.875 

.874 

.923 

.925 

1.025 

1.000 

1.020 

1.240 



Wine, champagne, 
** claret, 
*' port, 
** sherry. 



.997 
.994 

.997 



ELASTIC fluids: 

The measure of which is atmospheric air, 
at CAP. b. 30 in., il:j assumwl craviiy 1 ; on« 
cubic foot of which weii^hs 527.01 grains, ae 
.305 of a craiti per cubic inch. It i.s, al 
this tomiwralure and density, U) pure water 
at the maximum density, as .0012016 to I, 
or as 1 to 83.1.1. 



SI.MPLE OR ELEMENTARY GASF..^. 
.OOMi 

1.102.5 
.9700 



Hydrogen, 
OxyjTon, 
Nitrogen, . 
Fluorine, . 
('hlorine, 
Carhon, vapor of, 
(theoretically,) 

COMPOUND OASES. 

Ammoniacal, 

CarlK)nic acid, 
i ** oxide, . 

Carbureted hydrogen,. 

('hloro-carl)onic, 

C'yanogtMi, . 

Muriatic acid gas, 
• Nitrous acid gas, 
I Nitrous oxide ga.s, 
Olcfiant, . 

Phosphureted hydrogen, 
' Sulphureted ** 

Steam, 212° 

Smoke, of wood, 
' '* of coal, 
i Vapor, of water, 
, " of alcohol, 

of spirits turpentine. 



2.470 
\ .422 



.591 

1.525 

.763 

.559 

3.389 

1.818 

1.247 

3.176 

1.040 

.982 

1.185 

1.177 

.484 

.1)00 

.102 

.623 

1.613 

5.013 



WEIGnT PER BUSHEL — BARREL — GALLON, &C. 



35 



Weight 'per Bushel {even Winchester) of different Grains, Seeds, <!^c. 



Articles. 


lbs. 


Articles. 




IbB. 


Barley, (X. E. 47 lbs.) 


48 


Hemp seed. 


40 


Beans, 


64 


Oats, 




32 


Buckwheat, 


46 


Peas, 




64 


Blue-grass seed 


14 


Rve, 




56 


Corn, 


56 Salt, T. L, . 




80 


Cranberries, 


1 *' boiled, . 




56 


Clover seed. 


60 Timothy seed. 




46 


Dried Apples, 


22 Wheat, . 




60 


'' Peaches, 


33 Potatoes, h'p'd. 




60 


Flaxseed, (N. E. 52 lbs.) 


56 




Weight per Barrel {L 


egal or by Usage) of different Articles. 


Flour, . 


196 lbs. 


Cider, in Mass., 


32 gals. 


Boiled Salt, . . i 


>80 '' 


Soap, 


256 lbs. 


Beef, , . , i 


>00 '' 


Raisins, . 


112 *' 


Pork, . , . i 


^00 '' 


Anchovies, 


30 '' 


Pickled Fish, . . i 


JOO '' 


Lime, 




in > 
Massachusetts, ( 


30 gls. 


Ground Plaster, 




Hydraulic Cement, . 


300 *' 


A Gallon of Train 


il weighs . . " . 


7J lbs. 


A *' '' Molasse 


3, standard, (75 per cent.,) 


Ill '' 


A Puncheon of Prun 


es, .... 11 


20 ** 


A Firkin of Butter, (leoral.) 


56 '* 


A Keg of powder, 


. 


25 " 


A Hogshead of Salt ] 


s . . . . 


8 bush. 


A Perch of Stone = 


245 cubic feet. 




A Gallon of Alcohol, 


90 per cent., weighs . 


6.965 lbs. 


A '' '* Proof Spirits," '* 


7.732 *^ 


A *' " Wine, (ayerao;e,) " 


8.3 


A '' " Sperm 


il. 


7.33 *' 


A '' '' Whale < 


' p'f'd. 


7.71 *' 


A '' '' Olive * 


c n 


7.66 *' 


A *' '' Spirits Turpentine, *' 


7.31 '* 


A *' '' Camphei 


le, pure 




7.21 


<( 



Weight of Coals, cj-c, broken to the medium size, per Measure of 

Capacity, 

The average weight of Bituminous Coals, broken as above, is about 
62 per cent, that of a bulk of equal dimensions in the solid mass, or 



36 



ROPES AND CABLES. 



of the specific gravity of the article ; that of Anthracite is about 
57 per cent. 



Arerage weight 1 
per oubio foot. J 

Anthracite, . 
Bituminous, . 
Charcoal, of pine, . 
" of hard wood. 



Ibfl. W. Coal biuh«"l. 

54 1 Anthracite, 
50 1 Bituminous, 
18.6 ! Charcoal, hard wood, 
19.021 Coke, best, 



ib«. 
86 
80 
30 
32 



Practical Approximate Weight in Pounds of Variety Articles. 

Sand, dry, per cubic foot, .... 95 

Clay, compact, per cubic foot, . . . 135 

Granite, u u a ... 165 

Lime, quick, it u « ... 50 

Marble, nun ... 169 

Slate, " " " . . . 167 

Peat, hard, a u -a ... 83 

Seasoned Beech Wood, per cord, . . . 5616 

" Yellow Birch Wood, per cord, . . 4736 

" lied Maple AVood, " " . . 5040 

" '^ Oak Wood, ** « . . 6200 

" White Tine Wood, " " . 4264 

" Hickory Wood, " " . . 6960 

" ChesUiut Wood, " " . . 4880 

Meadow Hay, well settled, per cubic foot, 84 lbs., or 

240 cubic feet z= 2000 lbs., or 268^^ cubic feet 

z=z 1 lonjT ton 

Meadow llav, in larixc old stacks, per cubic foot, 

Clover Hay,* in settled bulk, " " " 

Corn on Cob, in crib, " " " 

" shelled, in bin, " " " 

Wheat, in bin, " " " 

Oats, in bin, " " " 

Potatoes, in bin, " " " 

Common Brick, 7| X 3| X 2 J in. " ]\I, . 

Front " 8 X 4i X 2^ in. " " . 



2 
5 

8 

4 



22 
45 

48 
251 
38| 
4500 
6185 



4 



ROPES AND CABLES. 

The STRENGTH of cords depends somewhat upon the fineness of 
the strands ; — damp cordage is stronger than dry, and untarred 
stonger than tarred ; but the latter is impervious to water and less 
elastic. 

Silk cords have three times the strength of those of fiax of 
equal circumference, and Manilla has about half that of hemp. 



WEIGHT AND STRENGTn OF IRON CHAINS. 



37 



Ropes made of iron wire are full three times stronger than those 
of hemp of equal circumference. 

White ropes are found to be most durable. The best qualities of 
hemp are — 1. 'pearl gray; 2. greenish; 3. yellow. A brown color, 
has less strength. 

The breaking weight T)f a good hemp rope is 6400 lbs. per square 
inch, but no cordage may be counted on with safety as capable of sus- 
taining a weight or strain above half that required to break it, and 
the weight of the rope itself should be included in the estimate. 

The reliable strength of a good hemp cable, in pounds, is usually 
estimated as equal to the square of its circumference in inches X by 
120. That of rope X 200. Thus, a cable of 9 inches in circumfer- 
ence may be relied on as having a sustaining power = 9 X 9 X ^"^^ 
= 9720 lbs. 

The weight, in pounds, of a cable laid rope, per linear foot = the 
square of its circumference in inches X .036, very nearly. 

The weight, in pounds, of a linear foot of manilla rope = the 
square of its circumference in inches X .03, very nearly. Thus, a 
manilla rope of three inches circumference weighs per linear foot 
3 X 3 X .03 =^ -iVd lbs., = 3yV f^et per lb. 

A good hemp rope stretches about ^, and its diameter is diminished 
about -^ before breaking*. 



WEIGHT AND STRENGTH OF IRON CHAINS. 



Diameter of 

AVire 

in Inches. 


Weight of 

1 Foot 
of Chain. 


Ereaking 
Wei.u-ht 
of Chain. 


Diameter of 

Wire 

in Inches. 


Weight of 

1 Foot 

of Chain. 


Breaking 
Weight 
of Cham. 




Lbs. 


lbs. 




lbs. 


lbs. 


3 
T6 


0.325 


2240 


1 


4.217 


26880 


i 


0.G5 


4256 


ii 


4.833 


32704 


A 


0.967 


6720 


f 


6.75 


38752 


t 


1.383 


9634 


if 


6.667 


45696 


T^ff 


1.767 


13216 


I 


7.5 


51744 


^- 


2.633 


17248 


it 


9.333 


58464 


^% 


3.333 


21728 


1 


10.817 


65632 



38 



COMPARATIVE WEIGHT OF METALS. 



Comparative Weight of Metals, Weight per Measure of Solidity , <!jrc. 





Specific • 


Ratio of 


Poun'l* 


in a Cabie 


Iron, wrought or rolled, 


GraTity. 


Companson 


Fool. 


Inrh. 


7.7b7 


1. 


486.65 


.28163 


Cast Iron, 


7.209 


.9258 


450.55 


.26073 


Steel, soft, rolled, . 


7.836 


1.0064 


489.75 


.28342 


Copper, pure, '' 


8.878 


1.1101 


554.83 


.32110 


Brciss, best, "■ 


8.G01 


1.1050 


537.75 


.3112 


Bronze, gun metal, . 


8.700 


1.1173 


543.75 


.31464 


Lead> .... 


11.352 


1.4579 


709.50 


.4106 



•TABLE, 

Exhilnting the Wright in pounds of One Foot in I/mgth of Wrought 
or Rolled Iroji of any size, {cross section,) from i inch to 12 inches. 



SQUARE BAR. 



Size 


Weight 


8it« 


Weight 


Site 


Weiehi 


8it« 1 Weight 


Inchei. 


Podnds. 


ln.hc». 

28 


Poiin«l«. 


Inches. 


round.. 


Inchei. 


Pounda. 


i 


.053 


19.066' 


48 


72.305 


71 


203.024 


i 


.211 


24 


21.120 


4J 


76.264 


8 


216.336 


1 


.475 


as 


23 292 


45 


80.333 


8i 


230.068 


i 


.84 5| 


2! 


25.560 


5 


84.480 
88.7ai 


H 


244.220 


1 


1.320 


2(S 


27.939 


5J 


B\ 


258.800 


i 


1.901 


3 


30.416 


5.t 


93.168 


9 


273.792 


I 


2.58S 


34 


33.010 


58 


97.657 


n 


289.220 


1 


3.380 


3t 


35.704 


54 


102.240 


!>4 


305.056 


U 


4.278 


38 


38.503 


58 


106.953 


9S 


321.332 


li 


5.280 


34 


41.408 


5i 


111.756 


10 


337.920 


11 


6.390 


38 


44.418, 


55 


116.671 


lOi 


355.136 


li 


7.604 


31 


47.534i 


6 


121.664 


104 


372.672 


11 


8.926 


35 


50.756 


6.1 


132.040 


103 


390.628 


IJ 


10.352 


4 


54.0841 


04 


142.816 


11 


408.960 


15 


11.883 


41 


57.517 


0} 


154.012 


m 


427.812 


3 


13.520 


4i 


61.055 


7 


165.632 


114 


447.024 


2i 


15.263 


41 


64.700 


7i 


177.672 


Hi 


466.684 


24 


17.112 


44 


68.448 


74 


190.136 


12 


486.656 



COMPARATIVE WEIGHT OF METALS. 39 

To determine the weight, in pounds, of one foot in length, or of any 
length, of a bar of any of the following metals of form prescribed, of 
any size, multiply the weight in pounds, of an equal length of square 
rolled iron of the same size, (see table of square rolled iron,) if the 
weight be sought of 

Iron, Round rolled, by 7854 

Steel, Square '' '' 1.0064 

'' Round '' " 7904 

Cast Iron, Square bar, ^* 9258 

'' " Round *' '' 7271 

Copper, Square rolled, '* 1.1401 

** Round '' "• 8951 

Brass, Square "• '' 1.105 

" Round '' '' 8679 

Bronze, Square bar, ** 1.1173 

** Round '' '* 8775 

Lead, Square '' " 1.4579 

Round " '' 1.145 . 

The weight of a bar of any metal, or other substance, of any given 
length, of ^ fiat form ^ (and any other form maybe included in the 
rule,) is readily obtained by multiplying its cubic contents (feet or 
inches) by the weight (pounds, ounces, or grains) of a cubic foot or 
inch of the article sought to be weighed ; that is — 

Length X hreadlh X thickness X weight per unit of measure. 

For the weight in pounds of a cubic foot or inch of diiferent metals, 
see ** Table of weights of metals per measure of solidity, &c.'' 

OR, FOR FLAT OR SQUARE BARS, 

Multiply the sectional area in inches by the length in feet, and that 
product, if the metal be 

Wrought Iron, by 3.3795 

Cast '' *' 3.1287 

Steel, " 3.4 

Example. — Required the weight of a bar of «teel, whose length is 
7 feet, breadth 2J inches, and thickness J of an inch. 

2.5 X .75 X 7 X 3.4 = 44.625 lbs. Ans. 

Example. — Required the weight of a cast iron beam, whose length 
is 14 feet, breadth 9 inches, and thickness IJ inch. 

14 X9 X 1.5 X 3.1287 = 591.32 lbs. Ans. 



40 



WEIGHT OF ROUND ROLLED IRON. 



TABLE, 

Exhibiting the weight in pounds of One Foot in Length of Round Rolled 
Iron of any diameter, from J inch to 12 inches. 



Diameter 


Weight 


Diara. in 


Weight 


Diam. In 


Weight 


Diam. io 


Weight 




in inches. 


in lbs. 

.041 


inches. 


in lbs. 


inches. 


in 11)S. 


inches. 


in lbs. 
150.466 




28 


14.075 


n 


56.788 


n 


i 


.165 


24 


16.688 


4| 


50.900 


8 


160.856 




i 


.373 


2| 


18.203 


4i 


68.094 


8t 


180.606 




h 


.663 


2J 


20.076 


6 


66.762 


8i 


101.808 




1 


1.043 


2i 


21.044 


6i 


60.731 


8i 


203.260 




i 


1.403 


3 


23.888 


H 


73.172 


9 


215.040 




J 


2.032 


8i 


25.026 


5! 


76.700 


^i 


227.152 




1 


2.654 


3.1 


28.040 


54 


80.304 


04 


230.6()O 




n 


3.360 


33 


80.240 


6| 


84.001 


9i 


252.376 




li 


4.172 


34 


82.512, 


S\ 


87.776 


10 


266.288 




1? 


5.010 


H 


84.886' 


H 


91.634 


101 


278.024 




14 


5.072 


35 


87.332 


6 


95.552 


104 


202.r,88 




u 


7.010 


35 


80.864i 


CI 


103.704 


lOj 


30r,.8()0 




ii 


8.128 


4 


42.4641 


04 


112.160 


11 


321.216 




n 


0.333 


4J 


45.174, 


GJ 


120.060 


Hi 


3;>6.004 




2 


10.616 


4t 


47.or,2' 


7 


130.048 


114 


351.104 




211 


11.088 


Al 


50.815, 


7i 


130.644 


Hi 


3<i6.536 




2i 


13.440 


^^4 


53.760[ 


7i 


140.328 


12 


382.208 





To find the xocight of an equilateral three-sided cast iron prism, 
width of side In inches^ X 1- 354 X length in feet = weight In lbs. 

Example. — A three-sided cast iron prism is 14 feet in length, and 
the width of each side is G inches ; required the weight of the prism. 

G-' X 1.354 X 14 = C82.4 lbs. Ans. 
To find the weight of an equilateral rectangular cast iron prism, 

width of side in inclics" X 3.12S X length In feet = weight In lbs. 

To find the weight of an equilateral five-sided cast iron prism, 

wi«ltli of side in inches" X 5.381 X length in f(.'et = weight in lbs. 

To find the weight of an equilateral six-sided cast iron prism, 

width of side in inches- X 8.128 X length in feet = weight in lbs. 
To find the locight of an equilateral eight-sided cast iron prism, 
width of side in inches' X 15.1 X length in feet = weight in Us. 

To find the weight of a cast iron cylinder, 

diameter in inches'" X 2.457 X length in feet = weight in lbs. 

In a quantity of cast iron weighing 125 lbs., how many cubic 
inches ? 

By tabular weight per cubic inch — 

125 -i- .26073 = 470.4 cubic inches. Ans. 



RELATING TO CAST IRON. 41 

Or, by tabular weight per cubic foot — 

450.55 : 1728 : : 125 : 479.4 cubic inches. Ans, 
How many cubic inches of copper will weigh as much as 479.4 
cubic inches of cast iron ? 

By tabular weight per cubic inch — 

.3211 : .26073 : : 479.4 : 389.27 cubic inches. Ans. 
. Or, by specific gravities — 

8.878 : 7.209 : : 479.4 : 389.27 cubic inches. Ans, 
Or, by tabular ratio of weight — 

.9258 
479.4 X 1J401 = 389.28. 

A cast ircn rectangular weight is to be constructed having a 
breadth of 4 inches and a thickness of 2 inches, and its weight is 
to be 18 lbs. ; what must be its length 1 

18 

4X2X.26073=^*^^'''^^^'- ^'^• 

A cast iron cylinder is to be 2 inches in diameter, and is to weigh 

6 lbs. ; what must be its length ? 

.26073 X .7854 =.2047 lb. = weight of 1 cyl. inch, then 
6 
2-y 2047 ~ "^'^""^ inches. Ans. 
A cast iron cylinder is to weigh 6 lbs., and its length is to bo 
7.327 inches ; what must be its diameter 1 

^\ 7.327 X .2047 ) = 2 inches. Ans. 

A cast iron weight, in the form of a prismoidj or the frustrum of 
a pyramid, or the frustrum of a cone, is to be constructed that wiU 
weigh 14 lbs., and the area of one of the bases is to be 16 inches, 
and that of the other 4 inches ; what must be the length of the 
weight ? 

yjx7= 8 -^"d 84^16 + 4 ^ 3 = 9.33, and 9.33 x^26073 

= 5.75 inches. Ans, ^n .-^wvic» 

Note, — For Rules in detail pertaining to the foregoing, see GEOMETRr, Mensuration 
of superficies — of solids. 

A model for a piece of casting, made of dry white pine, weighs 

7 lbs. ; what will the casting weigh, if made of common brass ? 
By specific gravities — 

.554 : 8.604 : : 7 : 108.71 lbs. Ans. 

Note. — As the specific gravity of the substance of which the model is composed must 
generally remain to some extent uncertain, calculations of this kind can only be relied on 
as ai^proximate. 

4* 



TABLE 

Exhibiting the Wcif/ht of One Foot in Lcnr/th of Flat, Rolled Iron; 
Breadth and Thickness in Inches , Wcif/ht in Pounds, 



IJr. and Th. 


Wei't. 


Br. and Th. 


Wei't. 


Br. and Th. 


WeiH. 


Br. and Th. 


Wei't. 


inch. 


lb a. 


inch. 


Ibn. 


inch. 


lbs. 


inch. 


Ibi. 


4 hy J 


.211 


n by I 


3.696 


\\ by 4 


2.957 


2J by 4 


8.591 




.422 


1 


4.224 


i 


3.696 


1 


4. 188' 




.634 


n 


4.752 


\ 


4.435 


i 




i I'y J 


.264 


n by i 


.581 


I 


5.175 


I 


• . 




.628 


i 


1.161 


1 


5.914 


1 


7. : i 


• 


,TX1 


I 


1.742 


li 


6.653 


li 


js_( r . 




1.056 


it 2.323 


li 


7.393 


1.1 


8.9771 


3 hy J 


.316 


g 2.904 


i« 


8.132 


Ijll 9.8741 




.633 


rj 


:Ms:. 


14 


8.871 


14 


10.772 




.950 


i 


l.< >(■,(; 


; 1.^ 


9.610 


2.1 by 1 


.960 




1.267 


1 


l.tWT 


IS by i 


.792 


d 


1.901 


1 


1.584 


\\ 


:..-Jli8 


i 


1.584 


i 


2.861 


J i^y 


.369 


l\ 


5.808 


i 


2.376 


4 


8.802 




.739 


11 by i 


.634 


4 


3.168 


» 


4.762 




1.108 


\ 


1.LV.7 


1 


8.960 


i 


6.708 


1 


1.478 




nl 


1 


4.752 


\ 


6.668 


i 


1.848 




1 


i 


6.544 


1 


7.604 


i 


2.218 




• ;s 


1 


6.836 


li 


8.664 


1 by.S 


.422 




. -' »2 


1) 


7.129 


U 


9.505 


4 


.845 


i 


4.435 


ij 


7.921 


1J10.1V> 


i 


1.267 


1 


5.069 


1 


8.713 


1411. 1'". 


4 


1.690 


in 


5.703 


14 


9.505 


18 12.356 




2.112 


l.i 


6.337 


111 


10.297 


13 13.807 


1 


2.534 


18 


6.970 


n 


11.089 


2| by \ 1.003 


I 


2.957 


1| by i 


.686 


2 by ^ .845 


i 2.00(^, 


n by i 


.475 


i 


1.373 


1 1.690 


1 3.010 


i 


.950 


8 


2.059 


I 


2.534 


41 4.«n;; 


j 


1.425 


4 


2.746 


4 


3.379 


|{ 5.(Hr, 


i 


1.901 


g 


3.432 


ft 


4.224 


% 6.01 9i 


& 


2.376 


5 


4.119 


i 


6.069 


I 


7.0-ja 


1 


2.851 


i 


4.805 


i 


6.914 


1 


8.<IL''. 


I 


3.326 


1 


5.492 


1 


6.759 


in 


9.(iL'.ti 


1 


3.802 


li 6.178 


IS 


7.604 


11 


10.(i:;-j 


li i^y i 


.528 


1.1 16.864 


li 


8.449 


liill.e::.;, 


i 


1.056 


1^7.551 


n 


9.294 


14|l2.n;v.»; 


s 


1.584 


1^8.237 


14 


10.138 


n 13.042, 


4 


2.112 


n by k\ 


.739 


2J by I 


.898 ' 


li 


l4.oi(;! 


1 


2.640 


i 


1.478 


h 


1.795 


2 


16.U52 


1 3.168 1 


1 


2.218 


I 


2.693 


24 by 4 


1.056 



WEIGHT OF FLAT, ROLLED IRON. 

TABLE. — Continued, 



43 



]Br.andTh 


Weight. 


Br. and Th 


Weight. 


Br. and Th 


Weight. 


Br. and Th 


Weight. 


inch. 


lbs. 


inch. 


lbs. 


inch. 


lbs. 


inch. 


lbs. 
20.594 


2iby i 


2.112 


2| by 11 


16.264 


3iby 1 


6.865 


3| by 1| 


I 


3.168 


n 


17.426 


il 8.238 


11 


'22.178 


h 


4.224 


2 


18.587 


I 


9.610 


n 


J23.762 




5.280 


2| 


19.749 


1 


10.983 


2 


125.347 


1 


6.336 


2. 


20.911 


1^112.356 


2i 


'28.515 


i 


7.393 


2J by J 


1.214 


li|13.729 


24 


31.683 


1 


8.449 


-.! 


2.429 


1115.102 


2| 


134.851 


IJ 


9.505 


f 


3.644 


li 16.475 


4 by i 


1.690 


li 


10.5G1 


4 


4.858 


11 


17.848 


i 


3.379 


liii.Gn 


1 


6.073 


Vi 


19.221 


4 


6.759 


li [12.073 


5 


7.287 


n 


20.594 


i 


10.139 


l|;l 3.729 


i 


8.502 


2 


21.967 


1 


13.518 


15 


14.785 


1 


9.716 


2i 


24.713 


li 


16.898 


U 


15.841 


lj;i0.931 


24 


27.459 


14 


20.277 


2 


16.898 


lil2.145 


34 by h 


1.478 


n 


23.657 


2S by J 


1.109 


lil3.3C0 


i 


2.957 


2 


27.036 


i 


2.218 


1414.574 


i 


4.436 


21 


30.416 


§ 


3.327 


IJ 


15.789 


4 


5.914 


24 


33.795 


h 


4.436 


li 


17.003 


t 


7.393 


2J 


37.175 


1 


5.545 


1U8.218 


1 


8.871 


3 


40.555 


i 


6.653 


2 


19.432 


i 


10.350 


3i 


43.934 


J 


7.762 


2J 


20.647 


1 


11.828 


4iby i 


1.795 


1 


8.871 


2i 


21.861 


n 


13.307 


i 


3.591 


IJ 


9.980 


3 by J 


1.267 


li 


14.785 


4 


7.181 


li 


11.089 


i 


2.535 


n 


16.264 


1 


10.772 


If 


12.198 




3.802 


H 


17.748 


1 


14.363 


li 


13.307 


A 


5.069 


ii 


19.221 


li 


17.954 


It, 


14.416 


i 


6.337 


n 


20.700 


14 


21.5441 


li 


15.525 


1 


7.604 


iji 


22.178 


li 


25.135 


li 


16.634 


J 


8.871 


2 1 


23.657 


2 


28.726 


2 


17.742 


1 


10.139 


21 


26.614 


2i 


32.317 


2J 


18.851 


n 


11.406 


24' 


29.571 


24 


35.908 


25 by J 


1.162 


u 


12.673 


2i 


32.528 


21 


39.498 


1 
4 


2.323 


if 


13.941 


3^^ by 1 


1.584 


3 


43.089 




3.485 


14 


15.208 


i 


3.168 


3i 


46.680 


* I 


4.647 


IS 


16.475 


1 


4.752 


34 


50.271 


1 


5.808 


15 


17.743 


4 


6.337 


44 by i 


3.802 


I 


6.970 


IJ 


19.010 


1 


7.921 


4 


7.604 


I 


8.132 


2i: 


20.277 


1 


9.505 


i 


11.406 


1 


9.294 


2 1 


22.812 


J 


11.089 


1 


15.208 


1^ 


10.455 


24; 


25.345 


1 


12.673 


li 


19.010 


li 


11.617 


H by h 


1.373 


n 


14.257 


14; 


22.812 


If 


12.779 


i 


2.746 


Id 


15.842 


i| 


26.614 


14 


13.940 




4.119 


If 


17.426 


2 


30.416 


l| 


15.102 


4 


5.492 


14 


19.010 


2i 


34.218 



44 



WEIGUT OF FLAT, ROLLED IRON. 
TABLE. — Continued. 



Br. and Th. 


Weight. 


Br. and Th. 


Weight. 


Br. and Th. 


Weight 


Br. and Th. 


Weight 


inch. 


lbs. 


inch. 


/.,. 


inch. 


lbs. 


inch. 


lbs. 


44 by 2i 


38.020 


4} by 3 


48.158 I 


51 by i 


13.307 


64 by 2 


37.175 


2J 


41.822 


3152.172 1 


1 


17.743 


24 


46.469 


8 


45.024 


84 


50.185 


11'22.178 


8 


55.762 


H 


40.420 


5 by i 


4.224 


1426.614 


6i by i 


4.858 


3i 


58.228 


i 


8.449 


n 


31.049 


4 


9.710 


n by i 


4.013 


i 


12.078 


2 


35.485 


i 


14.574 


4 


8.020 


1 


16.808 


21 


89.921 


1 


19.43;^ 


5 


12.040 


u 


21.122 


24 


44.350 


u 


24.290 


1 


10.053 


14 


25.347 1 


3 


53.228 


14|2y.l46j 


l.i 


20.00G 


n 


29.571 


54 by i 4.047 


n 


34.007 


14 


24.07'J 


2 


33.795 


4j 9.294 


2 


88.865 


n 


28.092 


2i 


38.020 


5 


13.941 


2i 


43.723 


2 


32.100 


24 


42.244 


1 


18.587 


24 


48.581 


2i 


30.119 


8 


4G.409 


1|' 23.234 


8 


58.297 


24 


40.132 


^i by \ 


4.430 


I427.88I 


6 by i 


5.0C9 


23 


44.145 


4 8.8711 


1^32.628 







WEIGHT OF METALS IN PLATE. 
The weight of a square foot o;ic inch thick of 



^lalleablo Iron 
Com. plate '' 
Cavst Iron 
Copper, wrought 

" com. plate 
Brass, plate, com. 
Zinc, cast, pure 

" sheet 
Lead, cast 



= 40.554 lbs. 

= 37.761 " 

=: 37.546 " 

= 46.240 " 

= 45.312 " 

= 42.812 " 

1=35.734 ** 

=1 37.448 " 

= 59.125 " 



And for any other thickness, greater or less, it is the same in pro- 
portion ; thus, a square foot of sheet copper ^\ of an inch thick 
= 46.24 -f- 16 = 2.89 lbs. And 5 square feet'at that thickness 
zn 2.89 X 5= 14.45 lbs., &c. So, too, 5 square feet at 2i inches 
thickness = 46.24 X 2.5 X 5 = 578 lbs. 



AMERICAN WIRE GAUGE. 



45 



THE AMERICAN WIRE GAUGE, 

The American Wire Gauge was prepared by Messrs. Brown and 
Sharp, manufacturers of machinists* tools, Providence, It I. It is 
graded upon geometrical principles, is rapidly becoming the stand- 
ard gauge with manufacturers of wire and plate in the United 
States, and cannot fail to supersede the use of the Birmingham 
Gauge in this country. 



TABLE 

Showing the Linear Measures represented hy Nos. American Wire 

Gauge and Birmingham Wire Gauge^ or the values of 

the Nos. in the United- States Standard Inch. 





American 


Birm. 1 




Araerican 


Birm. 




American 


Bim. ' 


American \ Birm. | 


No. 


(iauge. 


Gauge. 


No. 


Gauge. 


Gauge 


No. 


Oaoge. 


Oaoge. 


No. 


Oaoge. 


Gauge. 




Inch. 


Inch. 




Inch. 


Inch. 




Inch. 


Inch. 1 




Inch. 


Inch. 


0000 


.4G000 


.454 


8 


.12849 


.165 


19 


.03589 


.042 


30 


.01003.012 


000 


.40964 


.425 


9 


.11443 


.148 


20 .03196'.035 


31 


.00893'.010 


00 


.36480|.3-80' 


10 


.10189 


.134 


21 .02846L032 


32 


.00795'.009 





.3248G.340' 


11 


.09074 


.120 


22 .02535 .028 


33 


.00708.008 


1 


.28930.300 


12 


.08081 


.109 


i 23 1.022571.025 '34 


.00630.007 


2 


.25763/284 


13 


.07196 


.095 


24 1.02010 


.022 ; 35 


.0056i;.005 


3.22942'.259 


14 


.064081 .083 


25 1.01790 


.020 


36 


.00500.004 


4 


.204311.238 


15 


.05707 


.072 


26 L01594 


.018 


37 


.00445 




5 


.18194 


.220 


16 


.05082 


.065 


27 .01419 


.016' 


38 


.00396 




G 


.16202 


.203 


17 


.04526 


.058 


28 .01264 


.014' 


39 


.00353 




7 


.144281.180 


18 


.04030 


.049 


29 .01126 


.013,140 


.00314 





Thus the diameter or size of No. 4 wire, American gauge, is 
0.20431 of an inch; Birmingham gauge, 0.238 of an inch: so the 
THICKNESS of No. 4 platdy American gauge, is 0.20431 of an 
inch; Birmingham gauge, 0.238 of an inch; and so for the other 
Nos. on the gauges respectively. 



/ 



TABLE 

Showing the Number of Linear Feet in One Pounds Avolruupois^ of 
Differtnl Kinds of Wire ; Sizes or Diameters corre- 
sponding to Nos. Arnerican Wire-gauge. 



No. 


Iron. 


Copper. 


Brass. 


No. 


Iron, 


Copper. 


Brass. 




Fut. 


Feet, 


Feet. 


Feet. 


Feet. 


mtet. 


0000 


1.7834 


1.5616 


1.6552 


19 


293.00 


256.57 


271.94 


000 


2.2488 


1.9692 


2.0872 


20 


896.41 


847.12 


867.92 


00 


2.8356 


2.4830 


2.6318 


21 


465.83 


407.91 


432.35 





3.5757 


3.1811 


3.3187 


22 


587.85 


614.32 


645.13 


1 


4.5088 


3.9482 


4.1847 


23 


740.74 


648.63 


687.50 


2 


5.6854 


4.9785 


5.2768 


24 


984.03 


817.89 


866.90 


8 


7.1695 


6.2780 


6.6542 


25 


1177.7 


1031.3 


1093.0 


4 


9.0403 


7.9162 


8.3906| 


26 


1485.0 


1300.4 


1878.3 


5 


11.400 


9.9825 


10.581 


27 


1872.7 


1639.8 


1788.1 


6 


14.375 


12.588 


13.342 


28 


2861.4 


2067.8 


2191.7 


7 


18.127 


15.873 


16.824 


29 


2977.9 


2607.6 


2763.8 


8 


22.857 


20.015 


21.214 


30 


3754.8 


8287.9 


3484.9 


9 


28.819 


25.235 


26.748 


31 


4734.2 


4145.5 


4894.0 


10 


36.348 


81.828 


33.735 


32 


5970.6 


6221.2 


5541.4 


11 


45.829 


40.181 


42.535 


83 


7528.1 


6592.0 


6987.0 


12 


57.790 


50.604 


53.636 


34 


9495.6 


8314.9 


8813.1 


18 


72.949 


63.878 


67.706 


35 


11972 


10488 


11111 


14 


91.861 


80.439 


85.258 


36 


15094 


13217 


14009 


15 


115.86 


100.75 


107.53 


37 


19030 


16664 


17662 


16 


146.10 


127.94 


135.60 


38 


24003 


21018 


22278 


17 


184.26 


168.85 


171.02 


89 


30266 


26508 


28091 


18 


232.34 


208.45 


215.64 


1 40 


38176 


33342 


85482 



Note. — In this tabt.e the iron and copper employed are supposed to be 
nearly pure. The speciflc gravity of Die former was tulcen at 7.771 ; that of the 
latter, at 8,878. The spccilic gravity of the brass was taken at 8.370. 



\ 



WIRE AND WIRE GAUGES. 



47 



To find the number of feet in a pound of wire of any material not 
given in the table, of any size, American gauge, its specific 
gravity being known. 

Rule. — Multiply the number of feet in a pound of iron wire of 
the same size by 7.774, and divide the product by the specific grav- 
ity of the wire whose length is sought; or ordinarily, for steel wire, 
multiply the number of feet in a pound of iron wire of the same 
size by 0.991. 

To find the number of feet in a pound of wire of any given No., 
Birmingham gauge. 
Rule. — Multiply the number of feet in a pound of the same 
kind of wire, same No., American gauge, by the size, American 
gauge, and divide the product by the size, Birmingham gauge. 

Example. — In a pound , of copper wire No. 16, American 

fauge, there are 127.94 feet : how many feet are there of the same 
ind of wire, same No., Birmingham gauge ? 

(127.94 X .05082) -^ .065 i= 100.03. Ans. 

To find the weight of any given length of wire of any given No. or 
size, American gauge, or the length in any given weight, by help 
of the foregoing table. 
Example. — Required the weight of 600 feet of No. 18 iron 

wire. 

600 -r- 232.34 = 2.5822 lbs. = 2 lbs. 9 J oz., nearly. Ans, 

Example. — Required the length in feet of 2^ lbs. of No. 31 
brass wire. 

4394 X 2.5 zz: 10985. Ans, 

Characteristics of Alloys of Copper and Zinc — Brass, 



Parts by Weight. 


Specific 
Gravity. 


Color. 


Denomination. 


Copper. 


Zinc. 


83 
80 

49i 
33 


17 

20 

25J 

34 

50i 

67 


8.415 
8.448 
8.397 
8.299 
8.230 
8.284 


Yellowish Red. 

Pale yellow. 
FuU 

Deep " 


Bath Metal. 
Dutch Brass. 
Rolled Sheet Brass. 
English Sheet Brass. 
German Sheet Brass. 
Watchmaker's Brass. 



Note. — To alloys of copper and zinc, generally, there is added a small 
quantity of lead, which renders them the better adapted for turning, planing", 
or filing 3 and, for the same reason, to alloys of copper and tin, there is usually 
added a small quantity of zinc (see Alloys aijd Compositions), 



TABLE 



Showing the Weight of One Sfjuare Foot of Rolled MclaU^ thickness 
* corresponding to Nos.^ American Wire^auge, 



Thickness. 


Iron. 


Steel. 


Copper. 


Brass. 


Lead. 


Zinc. 


No, 


Pounds. 


Pounds, 


Pounds. 


Pounds, 


Pounds. 


Pounds. 


1 


lO.S-iO 


10.999 


13.109 


12.401 


17.102 


10.833 


2 


9.G611 


9.7953 


11.674 


11.043 


15.228 


9.6466 


3 


8.G032 


8.7227 


10.396 


9.8340 


13.562 


8.5:»'.:i 


4 


7.6G16 


7.7680 


9.2578 


8.7576 


12.078 


7.g:>oi 


5 


G.8228 


6.9175 


8.2442 


7.7988 


10.755 


6.8126 


6 


6.0758 


6.1601 


7.3416* 


6.9450 


9.5779 


6.0G67 


7 


5.4105 


5.4856 


6.5377 


6.1845 


8.5291 


5.4024 


8 


4.8184 


4.8853 


5.8222 


5.5077 


7.5957 


4.8112 


9 


4.2911 


4.3507 


5.1851 


4.9050 


6.7645 


4.2847 


10 


3.8209 


3.8740 


4.6169 


4.8675 


6.0283 


8.8151 


11 


8.4028 


3.4501 


4.1117 


3.8896 


5.8642 


8.8977 


12 


3.0303' 


3.0720 


3.6616 


8.4638 


4.7770 


8.0257 


13 


2.G985 


2.7360 


8.2607 


3.0845 


4.2539 


2.G1I34 


14 


2.4035 


2.4365 


2.9042 


2.74 73 


8.7889 


2.3!>99 


15 


2.1401 


2.1698 


2.5829 


2.4468 


8.3737 


2.1.S69 


IG 


1.9058 


1.9822 


2.3028 


2.1784 


8.0048 


l.9u2:i 


17 


1.6971 


1.7207 


2.0506 


1.9399 


2.6758 


1.6945 


18 


1.5114 


1.5324 


1.8263 


1.7276 


2.3826 


1.5091 


19 


1.3459 


1.3646 


1.6263 


1.5884 


2.1217 


1.3439 


20 


1.1085 


1.2152 


1.4482 


1.3700 


1.8893 


1.1967 


21 


1.0G73 


1.0821 


1.2897 


1.2300 


1.6768 


1.0657 


22 


.95051 


.96371 


U1485 


1.0865 


1.4984 


.94908 


23 


.84641 


.85815 


1.0227 


.96749 


1.8848 


.84514 


24 


.75375 


.76422 


.91078 


.86158 


1.1882 


.75262 


25 


.67125 


.68057 


.81109 


.7G728 


1.0582 


.67024 


2G 


.59775 


.60605 


.72228 


.68326 


.94229 


.59685 


27 


.53231 


.53970 


.64345 


JG0846 


.83918 


.53151 


28 


,47404 


.48062 


.57280 


.54185 


.74728 


.47333 


29 


.42214 


.42800 


.51009 


.48242 


.66546 


.42151 


^0 


.37594 


.38116 


.45426 


.42972 .59263{ 


.57538 



NoTK. — In calculatinpr the forrgoiiig tahlk, thi' Bpecific grnvitieu were 
taken as follow.'^ : viz., iron, 7.:^00; steel, 7.300; copper, 8.700; bradd, b::20', 
lead, 11.350; Zinc, 7.16i). 



TIN PLATES. 



49 



TIN PLATES. 



Brand 
Marks. 


Size of 
Sheets in 
Inches. 


No. of 
Sheets 
in Box. 


Net 1 
Weight 
in lbs. 


IC 


14X14 


200 


140 ! 


IC 


14X10 


225 


112 i| 


HC 


14X10 


225 


119 i 


HX 


14 X 10 


225 


147 


IX 


14X10 


225 


140 


IXX 


14X10 


225 


161 


IXXX 


14 X 10 


225 


182 j 


IXXXX 


14X10 


225 


203 I 


IX 


14X14 


200 


174 1 


IXX 


14X14 


200 


200 ! 


uc 


17X12i 


100 


105 


DX 


17X12^ 


100 


126 j 


DXX 


17X12^ 


100 


147 1 


DXXX 


17X12^ 


100 


168 i 


DXXXX 


17X12^ 


100 


189 ; 


SDC 


15X11 


200 


168 i 


SDX 


15X11 


200 


189 ' 



Size of I No. of Net 



Brand Marks. Sheets in Sheets 
I I 

I Inches, in Box. 



Weight 
in lbs 



SDXX 
SDXXX ' 
SDXXXX 
TT 

IC 

IX 

IXX 

IXXX 

IXXXX 

IC 

IX 

IXX 

IXXX 

IXXXX 

Ternes IC 

IX 



15X111 
15X11 
15X11J 
14 X 10, 
12X12; 
12X12, 
12x12: 
12X12 
12x12; 
20 X 14; 
20 X 14i 
20x14; 
20X14 
20 X 14; 
20 X 14 
20 X 14; 
I 



200 
200 
200 
225 
225 
225 
225 
225 
225 
112 
112 
112 
112 
112 
112 
112 



210 
231 
252 
112 
119 
147 
168 
189 
210 
112 
140 
161 
182 
203 
112 
140 



Note. — The above table includes all the regular sizes and qualities of 
tin plates, except " ivasters.^' Other sizes, such as 10 X 10, 11 X Hj 13 X 13, 
&c., of the ditterent brands, are often imported into the United States to 
order. 

Common English Sheet Iron, Nos. 10 to 28, Birmingham gauge, 
widths from 24 to S6 inches. 

li. G. Sheet Iron, Nos. 10 to 30, Birmingham gauge, widths from 
24 to 36 inches. 

American Puddled Sheet Iron, Xos. 22 to 28, Birmingham 
gauge, widths from 24 to 36 inches. 

Russia Sheet Iron, Nos. 16 to 8 inclusive, Russia gauge, sheets 
28 X 56 inches. 

Sheet Zinc, Nos. 16 to 8, Liege gauge, widths from 24 to 40 
inches ; length 84 inches. 

Copper Sheathing, 14 X 48 inches, 14 to 32 oz. (even numbers), 
per square foot. 

Yellow Metal, in sheets, 48 X 14; inches, 14 to 32 oz. (even num- 
bers), per square foot. 
5 



T.U3LE 

Showing the Capacity, in Wine Gallons, of Cylindrical Cans, oj 

different diameters , at One Inch depth. Diameter in Incites, 



Diam'r. 


Gallons. 


DiamV. 


OalloQS. 


DIamV. 


QaUodfl. 


DiamV. 


QaUon. 


inches. 




inche.n. 




inches. 




inches. 




6 


.1224 


12i 


.5102 


18i 


1.1G4 


241 


2.083 


61 


.1328 


124 


.5313 


18} 


1.195 


25 


2.125 


04 


.1437 


m 


.5527 


19 


1.227 


251 


2.167 


Oi 


.1549 


13 


.5740 


m 


1.200 


254 


2.211 


7 


.IGGG 


134 


.59(;9 


19i 


1.293 


251 


2.254 


7\ 


.1787 


m 


.G197 


I9i 


1.32G 


26 


2.298 


74 


.1913 


m 


.G428 


20 


1.3G0 


26i 


2.343 


7i 


.2042 


14 


.GGG4 


201 


1.394 


264 


2.388 


8 


.217G 


141 


.6904 


201 


1.429 


26} 


2.433 


8i 


.2314 


144 


.7149 


20J 


1.464 


27 


2.470 


84 


.2457 


1-U 


.7397 


21 


1.499 


27i 


2.524 


8i 


.2G03 


15 


.7G50 


21i 


1.535 


274 


2.571 
2.ft8 


9 


.2754 


m 


.7907 


214 


1.572 


271 


9.i 


.2909 


154 


.81G9 


21i 


1.608 


28 


2.666 


94 


.3009 


15J 


.8434 


22 


1.646 


281 


2.713 


9i 


.3233 


l(i 


.8704 


22.t 


1.683 


284 


2.762 


10 


.3400 


u->\ 


.8978 


22i 


1.721 


281 


2.810 


lOi 


.3572 


104 


.9257 


221 


1.760 


20 


2.850 


104 


.3749 


m 


.9539 


23 


1.799 


291 


2.909 


lOi 


.3929 


17 


.9826 


231 


1.837 


201 


3.009 


11 


.4114 


17t 


1.0120 


234 


1.877 


30 


3.060 


IH 


.4303 


174 


1.0410 


23J 


1.918 


304 


3.163 


114 


.4497 


m 


1.0710 


24 


1.958 


31 


3.264 


111 


.4G94 


18 


1.1020 


241 


1.999 


314 


3.374 


12 


.4896 


181 


1.1320 


^4 


2.041 


32 


3.482 



Applications of the foregoing table. 

Example. — A cylindrical can is 111 inches in diameter, and its 

depth is 18| inches ; required ite capacity. 

.4303 X 18? = 8 gaUona. Ans, 

Example. — The diameter of a can containing oil is 264 inches, and 
the oil is 144 inches in depth. How many gallons are there of the oil ? 

2.388 X 144 = 34. G gallons. Ans, 

Example. — A can is to be constructed that will hold just 36 Mil- 
Ions, and its diameter is to be 18 inches ; what must be its depth? 

36 -T- 1.102 =-321 inches. Ans. 



CAPACITY OF CYLINDEICAL CANS, ' 51 

Example. — A cylindrical can is to be constructed that shall have 
a depth of 15 inches and a capacity of just 5 gallons ; what must be 
its diameter ? 

5 -f. 15 = .3333 = capacity of can in gallons for each inch of depth ; 
and against .3333 gallon in the table, or the quantity in gallons 
nearest thereto, is 10 inches, the required, or nearest tabular diam- 
eter. Ans, 

Note. — The table is not intended to meet demands of the nature of the one contained in 
the last example, with accuracy, unless the fractional part of the diameter, if there be a 
fractional part, is i, ^ or ^ inch. As, however, the diamet^ opposite the tabular gallon 
nearest the one sought, even at its greatest possible remove, can be but about i inch from 
the diameter required, we can, by inspection, determine the diameter to be taken, or true 
answer to the inquiry, sufficiently near for practical purposes, be the fraction what it may. 
Or, to throw the demand into a mathematical formula : As the tabular gallon nearest the 
'One sought is to the diameter opposite, so is the tabular gallon required to the required 
diameter, nearly. Thus, in answer to the last query, 

.3400 : 10 : : 3333 : 9.8 iirches, the required or true diameter, nearly. 

For a mathematical formula strictly applicable to this question, see Gauginq 

Or, for a formula more strictly geometrical, we have 

.Capacity X 231 _,. 

jky ^-^ = diameter. 

^ Depth X -7864 ^"^"^^"^ 

The tme diameter, therefore, for the supposed can, is 

, J31><5_ ^^ 

^ 16 X -7864 ^^ 



52 * WEIGHT OF PIPES. 

WEIGHT OF PIPES. 

The weight of one foot in length of a pipe, ot any dinmet^ ? 
and thickness, may be ascertained by multiplying the square of it 
exterior diameter, in inches, by the weight of 12 rylindrical inches ol 
the material of which the pipe is composed, and by multiplying the 
square of its interior diameter, in inches, by the same factor and sub- 
tracting the product of the latter from that of the former, — the 
remainder or dilference will be the weight. This is evident from the 
fact that the process obtains the weight of two solid cylinders of e<jual 
length, (one foot,) the diameter of one being that of the pipe, and the 
other that of the vacancy, or bore. For very large pipes, the dimen- 
sions may be taken in feet, and the weight of a cylindrical foot of the 
material used as the factor, or multiplier, if desired. 

The weight of 12 cylindrical inches (length I foot, diameter 1 inch) 
of 

Malleable Iron = 2.0513 lbs. 

Cast Iron = 2. 1573 ** 

Copper, wrought, = 3.0317 *• 

Lead *' = 3.H()97 *' 

Cast Iron— 1 cyl. foot— = 353.80 ** 

Therefore — Example. — Required the weight of a copper pipe 
whose length is 5 feet, exterior diameter 31 inches, and interior 
diameter 3 inches. 

3i = -^ X Y = ^^-'^^^^-^ X 3.0317 = 32.022 + 
3 X 3 = y X 3.0317 = 27.285 -f 

Ans. 1.737 X 5 — 23.685 lbs. 

Example. — Required the weifiht of a cast iron pipe, whose length 
is 10 feet, exterior diameter 38 inches, and interior diameter 3 feet. 
38'^ X 2. 1573 — 30^ X 2.1573 = 303.08 X 10 = 3036.H U>s. Ans. 



Or, 38' — 30* = 148 X 2.4573 = 303.08 X 10 = 3030.8 lbs. Ans. 

Example. — Required the weight of a lead pipe, whose length is 
1200 feet, exterior diameter J of an inch, and interior diameter -j^ 
of an inch. 

t X 5 =-H = -05025, and -f^ X A = 7%V = -316406, and 
.705025 — .310400 = .449219 X 3.8097 X I'-^OO = 2080 lbs. Ans. 

Example. — The length of a cast-iron cylinder is I foot, ita 
exterior diameter is 12 inches, and its interior diameter 10 inches; 
required its weight. 

12^ — 10^ =-- 44 X 2.4573 = 108. 12 lbs. Ans. 
Or, 144 : 353.80 : : 44 : 108.12 lbs. Ans. 



WEIGHT OF PIPES. 



53 



The following Table exhibits the coefficients of weight, in pounds, of 
one foot in length, of various thicknesses, of different kinds of pipe, of 
any diameter whatever. 





Thickness 
in Inch«8. 


Wrought 
Iron. 


Copper. 


Lead. 






irV 


.332 


.379 


.484 






tV 


.664 


.758 


.9675 






^^2 


.995 


1.137 


1.451 






i 


1.327 


1.516 


1.935 






ife 


1.658 


1.894 


2.417 






T=V 


1.99 


2.274 


2.901. 






TfV 


2.323 


2.653 


3.386 






i 


2.654 


3.032 


3.87 






tV 


3.318 


3.79 


4.837 






t 


3.981 

CAST 


4.548 

IRON. 


5.805 








Thickness. 


Factor. 


Thickness. 


Factor. 


Thickness. | 


Factor. 


t\ 


1.842 


6.143 


l4^ 


12.287 




2.457 


i 


7.372 


li 


14.744 


1 


3.686 


i 


8.6 


If 


17.201 


h 


4.901 


1 


9.829 


2 


19.659 



To obtain the weight of pipes hy means of the above Table — 
Rule. — Multiply the diameter of the pipe, taken from the interior 
surface of the metal on the one side to the exterior surface on the 
opposite, (interior diameter -(- thickness,) in inches, by the number 
in the table under the respective metal'fe name, and opposite the 
thickness corresponding to that of the pipe — the product will be the 
weight, in pounds, of one foot in length of the pipe, and that product 
multiplied by the length of the pipe, in feet, will give the weight for 
any length required. 

Example. — Required the weight of a copper pipe whose length is 
5 feet, interior diameter and thickness 3 J inches, and thickness J of an 
inch. 



'H = 



3.125 X 1.516 X 5 = 23.687 lbs. Ans. 



Example. — Required the weight of a cast iron pipe, 10 feet in 
length, whose interior diameter is 3 feet, and whose thickness is 1 inch. 
36 -j- 1 = 37 X 9.829 X 10 = 3636.73 lbs. Ans. 
5* 



54 WEIGHT OF BALLS AND SHELLS. 

WEIGHT OF CAST IRON AND LEAD BALLS. 

To find the weight of a sphere or globe of any rnatcrial — 

Rule. — Multiply the cube of the diameter, in inches, or feel, bj 
the weight of a spherical inch or foot of the material. 
The weight of a spherical inch of 

Cast Iron . = .1305 lbs. 
Lead . . = .215 '' 

Therefore — Exa3iple. — Required the weight of a leaden baU 
whose diameter is \ of an inch. 

\ X \ X \ = -^\= .015625 X -215 = .00336 lb. Arts. 

Example. — Required the weight of a cast iron ball whose diameter 

is 8 inches. 

8^ X .1365 = 69.888 lbs. Ans. 

Example. — llow many leaden balls, liaving a diameter \ of an uich 
each, are there in a pound ? 

1 -r- .00330 = J-Ya'e'^^ = 298. Ans. 

Example. — What must be the diameter of a cast iroo ball, to 

weigh 69.888 lbs? 

09.888 -e- .1305 := ^512 = 8 inches. Ans. 

Example. — What must be the diameter of a leaden ball to equal 
in weight that of a cast iron ball, whose diameter is 8 inches? 
[Lead is to cast iron as .215 to .1365, as 1.575 to 1.] 
8^= 512 -^ 1.575 = ^325 = 6.875 inches. Ant. 



WEIGHT OF HOLLOW BALLS OR SHELLS. 
^ • 

The weight of a hollow ball is the weight of a solid ball of the 
same diameter, kss the weight of a solid ball whose diameter is that 
of the interior diameter of the shell. 

Example. — Rcquirotl the weight of a cast iron shell whose ex- 
terior diameter is 6 J inches, and interior diameter \\ inches. 

6 J =-2/- X -2^^ X ^i- = 241.11 X .1365 = 33.33 

4.1 =4.25' X .1365 = 10.48 

22.85 lbs. Ans. 

Or, If we multiply the difference of the cubes, in inches, of the two 
diameters — the exterior and interior — by the weight of a spherical 
inch, we shall obtain the same result. 

Example. — Required the weight of a cast iron shell whose ex- 
terior diameter is 10 inches and interior diameter 8 inches. 
103 _ Q3 y^ 1365 ::^ G6.612 lbs. Ans. 



ANALYSIS OF COALS. 



55 



ANALYSIS OF COALS. 



Pescription. 
Breckinridge, Ky., 
*' Albert," N. B., 
Chippenville, Pa. , 
Kanawha, " 
Pittsburg, *' 
Cannei, 
Newcastle^ 
Cumberland, 
Anthracite, a'v'g., 



Volatile Matter. 


Carbon. 


62.25 


29.10 


61.74 


32.14 


40. 8t) 




41.85 




32.95 




35.28 


64.72 


24.72 


75.28 


18.40 


80. 


3.43 


89.46 



A&. 

8.65 
6.12 



1.60 
7.11 



Woods of most descriptions vary little from 80 per cent, volatile 
matter, and 20 per cent, charcoal. 

Table — Exhibiting the TVcights, Evaporative Powers, ^c,^ of Fuels^ 
from Report of Professor Walter R. Johnson. 



\ 




Weiu'lit 


Lbs. ul Water 
at 212 fle^rees 


Lbs. of Water 


Weight of 


Designation of Fuel. 


Specific 
Griv- 


per 
Cubic 


converted into 
Steam by 1 


rti 212 degrees 
conTeried^into 


C4iDkers 
from 100 lb«. 




ily. 


I-'ool. 


Cubic Foot of 


Steam by 1 lb. 


of Coal. 








Fuel. 


of Fuel. 




Anthracite Coals. 










Beaver Meadow, No. 3 


I.GIO 


54.93 


526.5 


9.21 


1.01 


Beaver Meadow, No. 5 


1.554 


56.19 


572.9 


9,88 


.60 


Forest Improvement 


1.477 


53.66 


577.3 


10.06 


.81 


Lackawanna 


1.421 


48.89 


493.0 


9.79 


1.24 


Lehigh 


1.590 


55.32 


515.4 


5.93 


1.08 


! Peach Mountain 


1.464 


53.79 


581.3 


10.11 


3.03 


Bituminous Coals. 












Blossburgh 


1.324 


53.05 


522,6 


9.72 


3.40 


Cannclton, la. 


1.273 


47.65 


360.0 


7.34 


1.64 


Clover Hill 


1.285 


45.49 


359.3 


7.67 


3.86 


Cumberland, average. 


1.325 


53.60 


552.8 


10.07 


3.33 


Liverpool 


1.262 


47.88 


411,2 


7.84 


1.86 


Midlothian 


1.294 


54.04 


461.6 


8.29 


8.82 


Newcastle 


1.257 


50.82 


453.9 


8.66 


3.14 


Pictou 


1,318 


49.25 


478.7 


8.41 


6.13 


Pittsburgh 


1.252 


46.81 


384.1 


8.20 


.94 


Scotch 


1.519 


51.09 


369.1 


6.95 


5.63 


Sydney 


1.338 


47.44 


386.1 


7.99 


2.25 


Coke. 












Cumberland 




31.57 


284.0 


8.99 


3.55 


Midlothian 




32.70 


282.5 


8.63 


10.51 


Natural Virginia 


1.323 


46.64 


407.9 


8.47 


5.31 


Wood. 












Dry Pine Wood 




21.01 


98.6 


4.69 





56 MENSURATION OP LUMBER. 



MENSURATION OF LUMBER. 

To find the contents of a board. 

Rule. — Multiply the length in feet hy the width in inches, an<l 
divide the product by 12 ; the quotient will bo the contcnta in squaro 
feet. 

Example. — A board is IG feet long and 10 inches wide; bow 
many square feet doc3 it contain ? 

10 X 10 =« 100 -^ 12 = i:', .'j. Ans. 

To find the contents of a plank ^ joist, or stick of square timber. 

Rule. — Multiply the pru<luct of the depth and width in inches by 
the length in feet, and divide the laat prouiict l»y 1- ; tl»o quotient \s 
the contents in feet, board measure. 

Example. — A jjjist is IG feet long, 5 inrhoh* wkic, and 2.^ inches 
thick ; how many feet does it contain, board measure ? 

5 X 2.5 X IG -T- 12 = IG/j. Ans, 

To find the solidity of a plank, joist, or stick of square timber. 

Rule. — Multiply Uie prcnluct of the depth and width in inches 
by the length in feet, and divide the hist jiroduct by 144 ; the quo- 
tient will be the cuntcnts in cubic feet. 

ExAMPi^. — A stick of timlwr is 10 liy G inches, and 14 feet in 
length ; what is its solidity ? 

10 X G =. 130 X 14 =» 840 -r- 144 = 5 J feet. Ans, 

Note. — If aboanl, nlnnk, or ^oist !« narrower nt one end than the other, 
add the two ends toLM'tiior ami divido the sum by 2; the nuotlrnt will \h' the 
mean widtli. And if a slick of s<iiiared tiniUr, wliose solitlity id rr.juired, \9 
narrower at one end than the oth< r (A -^a-^-*^/ Tin ; -i- a = mean area. A and 
a being the areas of the ends. 

To measure round timber. 

Rule (in general practice.^ — Multiply the len^rth, in feet, by 
die square of.} the ;:^irt, in inches, taken about \ the distance from 
the larger end, and divide the product by 144 ; the quotient is con- 
sidered the contents in cubic feet. For a strictly correct rule for 
measuring round timber, see Mensuration ok Solids — Frustum oT n 
Cone. 

Example. — A stick of round timber is 40 feet in length, anJ 
girts 88 inches ; what is its solidity ? 

884-4 = 22 X22 = 484 X 40 = 193G0 -r- 144 = 134.44 cub. ft. Ans 



MENSCRATIOJ; OF LUMBER, 



57 



The following TABLE is intended to facilitate the measuring of Round 
Timber, and is predicated upon (he foregoing Rule. 



i Girt in 


Area in 


i Girt iu 


Area in 


i Girt in 


Area in 


i Girt in 


Area in 


Inches. 


Feet. 


Inches. 


Feet. 


Inches. 


Feet. 


Inckes. 


Feet. 


6 


.25 


12 


1, 


18 


2,25 


24 


4. 


6.t 


.272 


m 


1,042 


181 


2.313 


241 


4.084 


64 


.204 


m 


1.085 


184 


2.376 


244 


4.168 


6| 


.317 


m 


1,120 


18| 


2,442 


24| 


4.254 


7 


.34 


13 


1.174 


19 


2.506 


25 


4.34 


7.i 


.304 


131 


1,210 


191 


2.574 


2H 


4.428 


7A 


.30 


13.1 


1.265 


194 


2,64 


254 


4.516 


7;i 


.417 


mi 


1.313 


19i 


2.700 


25i 


4.605 


8 


,444 


u 


1.361 


20 


2,777 


26 


4.604 


8i 


.472 


141 


1.41 


201 


2,808 


26i 


4.785 


H 


.501 


Ui 


1,46 


204 


2.017 


264 


4.876 


8| 


-.531 


14|' 


1.511 


20i 


2,00 


26i 


4.060 


9 


,562 


15 


1.562 


21 


3,062 


27 


5.062 


91 


.504 


151 


1.615 


211 


3,136 


27i 


5.158 


9i 


.620 


154 


1.6G8 


214 


3,200 


274 


5.252 


n 


650 


m 


1,722 


21i 


3.285 


27i 


5.348 


10 


.604 


10 


1.777 


22 


3,362 


28 


5.444 


m 


.73 


161 


1.833 


221 


3,438 


28i 


5.542 


m 


.766 


lO.J 


1.80 


22.i 


3,516 


284 


5.64 


lOj 


.803 


ir.j 


1.048 


225 


3.508 


28i 


5.74 


11 


.84 


17 


2.006 


23 


3.673 


20 


5.84 


Hi 


.878 


171 


2,066 


23t 


3,754 


20i 


5.041 


m 


.018 


174 


2,126 


23.J 


3.835 


204 


6.044 


in 


.050 


17J 


2,187 


23| 


3.017 


30 


6.25 



To find the snlidity of a log by help of the preceding t^ujle. 

Rule. — Multiply the tahular area opposite the corresponding 
\ gh't, l\v the length of the log in feet, and the product will be the 
solidity in feet. 

Example. — The \ prt of a loo; is 22 inches, and the length of the 
log is 40 feet ; required the solidity of the log. 

3.362 X 40 = 134,48 cubic feet. ^715. 

, Note. — Though custom has established, in a very general way, the preceding method 
as that whereby to measure round timber, and holds, in most instances, the solidity to be 
that which the method will give, there seems, if the object sought be the real solidity 
of the stick, neither accuracy, justice, nor certainty, in the practice. 

Thus, in tlie preceding example, the stick was supposed to be 40 feet in length, and 8S 
inches in circumference at ^ the distance from the larger end, and was found, by the 
method, to contain 134.44 cubic feet : now 88 -^ 3.1416 = 28 inches, = the diameter at i 
the distance from the gi'eater base, and retaining this diameter and the length, we may 



58 , MENSURATION OF LUMBER. 

Buppose, with sufScIent liberality, antl without being f:ir from the pcncral run nf Bnch 
sticks, the diameter at the greater base to be 30 inches, and that of the less to be .H 

inches, and — 
By a correct rule the stick contains — 

30 X 24 = 720 + 12 = 732 X. 7854X40 = 22906 4-144 = 159.7 cubic feet, or 19 per 

cent, more than given by the method under consideration -, and we nee<i ). .r.iiv ^.1.1 t^r^t 
the nearer the stick approaches to the figure of a cyUnder, the wider will 
between the truth and the result obtainetl by the method referretl to. T! i. j 

Btick a cylinder, 2S inches in diameter, and 40 feet in length ; and we h*.-., iv m • Kuia- 
cious rule, as above, 134.44 cubic feet *, and — 
By a correct method, we have — 

28- X. 7854X40 = 24630 -7-144 = 171 cubic feet, or over 27 per cent, more than fur- 
nished by the erroneous mode of practice. 

Again : suppose the stick in the form of a cone, 30 Inches at the base, and tapering to A 
point at 150 feet in length j and we have, by a correct rule — 

30--^ 3 = 300 X -7851X150 = 35343 -7- 144 =245.44 cubic feet-, and by the ordinary 
method of gauging, or the aforementioned practice, we have — 

20 X 3.1410 = 62.8;V2 4-4 = 15.70S- X l'''0 = 37011.19 +144 = 257 cubic fcet,or nearly 
4i{ i»er cent, more tlian *' ■ ' ' •■'■-■•.•i. 

In short, without taki for the thickness of the kirk, that may 

be supj)osod to be o!i tl. -rrect only when the stick laiKTS at the 

rate of 51 inc]i<s il!;iiijrii.r i< r ( .icli lu f ■ « t in Kngth, or o¥^r i Inch diameter to each fool 
in length of the stLk. 

If, iutwever, wc supiiose the slick as before, (30 inches at the grecOter Iwwe, 24 Inches at 
the smalkr, and 40 fcrt in l«Mi:/lh,) and suppose the Iwrk uimn it to be 1 inch tldck, wo 
shall have, by the u«iual meth<Kl, 134.44 cubic feet, as before. And, exclusive of the bark, 
by a corrt'ct metlunl, we shall have. 

3U — 2 X 24 — 2 = 610 + 12 = 62S X .7854 X 40 = 19720 + 144 = 137 cubic feel, or 
only about 2 per cent, more than that furnii«hcd us by the u^ual practice. 

Tho followinf]; simple rule for measurinp; round timber ia suffi- 
ciently correct fl)r most practical purposes : — 

Rule. — Multiply the square ofone-lifth of tho mean prt, (cxclu- 
eivo of bark,) in inches, hy twice tho Icn^^th of tho stick in feet, and 
divide the product by 144 ; the quotient will bo the solidity in feet. 

To find the solidity of the greatest rectangular stick that may he cut 
from a given loa'. or from a stick of round iimlicr of 'jinn dimrn- 
sions. 

Rule. — Mulripiy the square of the mean diam«»t<'r of tho lon^, in 
inches, l)y half the length of the log, in feet, and divide the product 
by 144. 

Example. — The diameter (exclusive of bark) of the greater baso 
of a stick of round timber is 30 inches, an<l that of tho loss base is 
24 inches, and the stick is 40 feet in length ; required the solidity 
of the greatest rectangular stick that may be cut from it. 

30X24 + ^30 — 24)- = 732 = square of mean diameter,* and 

732X 20 = 14«j40 -7-144 = 101? cubic feet. Ans. 

* Except in the case of a cylinder, there is a difference betwixt the mean diameter nf a 
solid having circular bases, and the middle diameter of that solitl. The mean diameter 
reduces the solil to a cyUnder j the midiUe diameter ia the diameter midway l>ctweei\ tlie 
two bases. 



MENSURATION OF LUMBER. 59 

Note. — The foregoing stick will make — 

14640 -^ 16 = 915 feet of square-edged Iwards 1 inch thick \ 
Or, 101§X9=915. 

To find the solidity of the greatest square stick that may he cut from a 
given log, or from a stick of round timber of given dimensions. 

Rule. — Multiply the square of the diameter of the less end of the 
log, in inches, by half the length of the log, in feet, and divide 
the product by 144. 

Example. — The preceding supposed log will make a square stick 
containing — 

242 X -4^ = 1152 -^ 144 = 80 cubic feet. 

Diameter multiplied by .7071 = side of inscribed square. 

To find the contents^ in Board Measure, of a log, no allowance being 
made for wane or saw-chip. 

Rule. — Multiply the square of the mean diameter, in inches, by 
the length in feet, and divide the product by 15.28. 

Or, Multiply the square of the mean diameter in inches, by the 
length in feet, and that product by .7854, and divide the last prod- 
uct by 12. 

The cubic contents of a log multiplied by 12, equal the contents 
of the log, board measure. 

The convex surface of a Frustum of a Cone = (C -|- c) X -J slant 
length ; C being the circumference of the greater base, and c the 
circumference of the less. 



60 GAUGING. 

GAUGING. 

Rules for finding the capacity in gallons or bushels of diffcreni 
shaped Cisterns J Bins, Casks , (^c, and also, by way of examples, for 
constructing them to given capacities. 

Rule — 1. When the vessel is rect an giilar. IMulti ply the interior 
length, breadth, and depth, in feet togeihcr, and the product by the 
capacity of a cubic foot, in gallons or bushels, as desired for its 
capacity. 

Rule — 2. When the vessel is cylindrical. Multiply the square of 
its interior diameter in feet, by its interior depth in feet, and the prod- 
uct by the capacity of a cylindrical foot in gallons or bushels, as 
desired for its capacity. 

Rule — 3. \\ hen the vessel is a rhombus or rhomboid. Multiply. 
Hs interior length, in feet, its right-anfrular breath in feet, and its 
depth in feet together, and the product by the capacity of a cubic foot 
in the special measure desired for ita capacity. 

Rule — 4. When the vessel is a frustum of a cemc — a round vessel 
larger at one end than the other, whose bases are planes. Multiply 
the interior diameter of the two ends together, in feel, add i the 
•quare of their dilTercnce in feet to the product, multiply the sum by 
the perpendicular depth of the ressel in feet, and that product by tho 
capacity of a cylindrical foot in the unit of measure desired for ita 
capacity. 

Rule — 5. When the vessel is a prismoid or the frustum of any 
regular pt/rarnid. To the square root of the product of the areas of its 
ends in Icet, add the areas of its ends in fwt, multijjly the sum by 
\ its per])endicular depth in feet, and that product by the capacity of 
a cubic foot in gallons or bushels, as d(*j^ired for its capacity. 

If it is found more convenient to take the dimensions in inches, do 
so ; proceed as directed for feet, divide the product !)y 172H, and mul- 
tiply the quotient by the capacity of the respective foot as directed. 
Or, multiply tlie capacity in inches by the capacity of the respective 
inch in gallons or bushels; — by the quotient obtained by dividing the 
capacity of the respective foot in gallons or bushels by 1728 — fc^r 
the contents. 

Rule — 6. When the vessel is a barrel, hogshead, pipe, 4'C, Mul- 
tiply the dilTerenco in inchies l>etween the bung diameter and head 
diameter, (interior,) if the staves be 

much cun'ed, . by .7 1 

medium curved, . hy .05 I ^^ -„ 

straighter than medium, by .0 f^^ ^^ ^'^' 
nearly straight, . by ..55 J 

and add the product to the head diameter, taken in inches ; then mul- 
tiply the square of the sum by the length of the cask in inches, and 
divide the product by the capacity in cylindrical inches of a gallon oi 



aAUGINQ. 61 

bushel as desired for the contents. Or, divide the contents in cylin- 
drical inches, as above found, by 1728, and multiply the quotient by 
the capacity of a cylindrical foot in gallons or bushels as desired for 
its contents. Or, multiply the capacity in cylindrical inches by the 
capacity of a cylindrical inch, in gallons or bushels, as desired, — 
that is, by the quotient obtained by dividing the capacity of a cylin- 
drical foot in gallons or bushels, by 1728, for the contents. 
The capacity of a 



CUB re FOOT = 

7.4805 Winchester wine gallons. 

6.1276 i\lc 

6.2321 Imperial '' 

,80350 Winchester bushel. 

.02888 " heaped '' 

.64285 " li even " 

,779 Imperial '' '' 



CYLINDRICAL FOOT = 

5.8751 AVinchester wine gallons. 
4.8120 Ale '' 

4.8947 Imperial " 

.63111 Winchester bushel. 

.49391 '' heaped ** 

.50489 '' lieven " 

.61183 Imperial '' 



Example. — Required the capacity in Winchester bushels of a 
rectangular bin, whose interior length is 12 feet, breadth 6 feet, and 
depth 5 feet. 

12 X G X 5 X .B035 = 289.26 bushels. Ans. 
Example. — Required the capacity in AVinchester wine gallons of 
a cyhndrical can, whosrj ijiterior diumeter is 18 inches, and depth 3 
feet. 

18 X 18 X 30X 5.875 ^ 1728 = 39.66 gallons. Ans. 
Or, 1.5 X 1.5 X 3 X 5.875 = 39.66 gallons. Ans. 

Or, 18 X 18 X 36 X -0034 = 39.66 gallons. Ans. 

Example, — IIow many Winchester bushels in 39.66 wine gal- 
lons^ 

39.66 X .10742 = 4.26 bushels. Ans. 

Example. — How many wine gallons in 4.26 Winchester bushels? 
4.20 X 9.3092 = 39.66 gallons. Ans, 

ExAMPLi:. — How many wine gallons will a cistern in the form of 
a frustum of a cone hold, having the interior diameter of one of its 
ends 6 feet, and that at the other 8 feet, and its perpendicular depth 
9 feet? 

8 — 6 = 2, and 2- -^ 3 = 1.333 = J square of dif. of diameters, and 
6X8 + 1.333 = 49.333 X 9 X 5.8751 = 2608.55 gals. Ans. 

Or, 6 X 8 + 8^' + 6^' = 148 X f X 5.8751 = 2608.55 gals. Ans. 
Or, (8' — 6'' ) -1- (8 — 6) = 148 X f X 5.8751 == 2608.55 gals. Ans. 
Or, 90 — 72 == 24 and (24-^ -~ 3) = 192, and 

90 X 72 + 192 = 7104 X108 X .0034 = 2608.55 gals. Ans. 
6 



62 0AUGI5G. 

ExACPLE. — What is the capacity in Winchester bushels of a cis» 
tern whose form is prismoid, the dimensions (interior) of one end 
being 8 by 6 feet, of the other 4 by 3 feet, and its perpendicular 
depth 12 feet ? 

8 X 6 = 48 = area of one end, and 4X3 = 12= area of the other 
end ; then — 

48 X 12 = V 57G = 24 + 48 + 12 = 84 X-^ X .80356 = 270 bush- 
els. Ans. 

Or, (8 + 4) -T- 2 = 6, and (G + 3) -^ 2 =«4.5 =» mean sectional areas 
of ends, and 

G X 4.5 X*4 = 4 area of mean perimeter, then 
8XG + 4x3 + Gx4.5x4 = 168XYX.80356 = 270bus. Ans, 

Example. — What must be the depth of a roctanji^idar bin whoso 
length is 12 feet, and ])readth G feet, to liold 2.S1).2G Imshels? 
289.26 -7- (12 X G X .80356) = 5 feet. -4;^. 

BKAifPLE. — A cylindrical can, whose depth is to lyo 36 inches, is 
required to Ix) made that will hold 40 g:ulons ; what must bo tho 
diameter of tho can ? 

40 — (3 X 5.8751) = V2.27 = 1.50G feet. Am. 
Or, 40 -^ (36 X .0034) = V326.8 = 18.07 inches. Ans. 

Example. — A cylindrical can, whose interior diameter is to bo 13 
inches, is required that will hold 40 [rallons ; what must bo tho 
interior depth of tho can ? 

40 -T- (18- X .0034) = ;io.:U inchcH. Ans. 
Or, 40 -f- (1.52 X 5.8751) = 3.026 feet. Ans, 

Example. — A cistern is to be built in tlic form of a frustum of a 
cone, that will hold 1800 gallons, and the diameter of one of its 
ends is to be 5 feet, and that of the other 7i feet ; what must be tho 
depth ? 

7.5 — 5 = 2.5, and 2.5- — 3 = 2.0833 = J square of difference of 
diameter, and 

1800 -^ (7.5 X 5 + 2.0833) X 5.8751 = 7.74 feet. Ans. 
/7.5X5 + 7.5-' + 52 \ 

Or, 1800 -r-l 3 X 5.8751 )= 7.74 feet. Ans. 

Example. — Tho form, capacity, depth, and diameter of one end 
being determined on, and being as above, what must be the diameter 

of the other end ? 

c 
-TT- — %d^ = !/j c being the solidity in cylindrical measurement, h 



GAUGING. 63 

the depth, d the diameter of the given end or base, and y a quantity 
the square root of which is the sum of the required base and half the 
given base ; then 

1800 -i- 5.8751 = 306.378 = solidity in cylindrical feet, and 
306.378 -^ ^^4 = 118.75 __ (52 ^ 4 ^ ^ ^^^^ = 10 — J = 7.5 
feet. Ans, 

Example. — A measure is to be built in the form of a frustum of a 
cone, that will hold exactly 1 wine gallon, and the diameter of 
one of its ends is to be 4 inches, and that of the other 6 inches ; 
what must be its depth ? 

1 -7- (6 X 4 + IJ) X .0034 = 11.61 inches. Ans. 

231 6 X 4 + 6^ + 42 
Or, ^054 -7- ~y = 11.61 inches. Arts, 

Example. — A measure in the form of a frustum of a cone holds 1 
wine gallon ; the diameter of one of its ends is 6 inches, and its 
depth is 11'61 inches ; what is the diameter of the other end ? 

m% = 294.1176 ^ i-i^6_i = 76 _ (G^ -u f) = V49 = 7 - f 

= 4 inches. Ans, 



CASK GAUGING. 

Cask-gauging, in a general sense, is a practical art, rather 
than a scientific achievement or problem, and makes no pretensions 
to strict accurracy with regard to the conclusions arrived at. The 
aim is, by means of a few satisfactory measurements taken of the 
outside, and an estimate of the probable mean thickness of the ma- 
terial of which the cask is composed (of which there must always 
remain some doubt), or by means of a few measurements taken of 
the inside, to determine, 1st, the capacity of the cask, and, 2d, the 
ullage, or capacity of the occupied or unoccupied space in a cask 
but "partly full. And the Rule (Rule 6, page 60^, which re- 
duces the supposed cask, or cask of supposed curvature, to a cylin- 
der, is as practically correct for the capacity of ordinary casks, as 
any rule, or set of rules, tlfat can be offered for general purposes. 

Casks have no fixed form of their own, to which they severally 
and collectively correspond, nor are they in any considerable degree 
in conformity with any regular geometrical figure. 

Some casks — a few — those having their staves much curved 
throughout their entire length, are nearest in keeping with the 
middle frustum of a spheroid ; others, slightly less curved than 
the preceding, correspond in a considerable degree to the middle 



Ui GAUGING. 

frustum of a parabolic spindle; others, agjiin — those havini; very 
little longitudinul curvature of stave to tlieir semi-lengths — are 
nearly in keeping with the ajual frustums of a 'paraboloid ; and others 
— a very few — those whose staves are straight from tke bung diam- 
eter to the heads, or equal to that form, are in accordance with the 
cijual frustums of a cone. 

The gauging rod, \yhich is intended to be con*cct for casks of the 
most common form, gives for all casks, as may l)e seen in one of the 
following Exami'Lp:s, a solidity slightly greater (about 2.i per cent.) 
than would be obtained by supposing the cask in conformity wiili 
the third figure above alluded to. 

The Ki LK for finding the contents of a cask, by four dimensions. 
hereafter to be given, is intended as a general Rule for all casks, 
and, wh(;n tlic diameter midway between the bung and liead can I" 
accurately ascertained, will lead to a very close Approach to tli. 
truth. 

From the length of a cask, taken from outside to outside of tin 
heads, with callipers, it is usual to deduct from 1 to 2 inches, to c<>r 
respond with the thickness of the heads, according to the size of tiio 
wasK, and the remainder is taken as the length of the interior. 

To the diameter of each head, tiiken extermilly, from i inch to 
-fj inch should be added for common-sized barrels, -i*j inch for 40 gal- 
lon casks, and from i inch to ^{^ inch for larger casks, to correspond 
with the interior diameters of the heads. 

If tiie staves are of uniform thickness, any sectional diameter of a 
cask may bo nearly or quite ascertained, by dividing the circumfer- 
ence at tiiat j)lace by 3.U1G, and subtracting twice the thickness of 
tlie stave from tlie quotient. 

For obtaining the diagonal of a cask by mathematical process, — 
the interior Imigth, tto. »fcc. — see Rubs, l)clow. 

In the following forinuhus I) denotes the bung diameter, d the 
head diameter, and / the length of the cask. 

The solidity of any cask is equal to its length multiplied by the 
square of its moan diain 't<:r multiplied by .7S')4. 



To cahuiait in> (oiu'iitsofacaskfrumj 



1 1)11 r (iinv nsions. 



Kile. — To the square of the bung dtameter add the square of 
the head diameter, and the square of double the diameter midway 
between the bung and head, and multiply tlie sum by ^ the length 
of the cask, for its cylindrical contents ; the product multiplied by 
.0034 expresses tlie contents in wine gallons. 

Ex.\MrLE. — The length of the cask is 40 inches, its bung diameter 
28 inches, head diameter 20 inches, and the diameter midway bc^ 



GAUGING. 65 

tween the bung and head is 25. G inches ; how many gallons' capacity 
has the cask? 

202 + 282 + 25.6 X 2^ = 3805.44 X "V" X .0034 = 86.26 gals. Ans. 

(D2 + 6/2 + 2rn ) X i ^ X -7854 = cubic contents. 

D2 + rf2 + 2^' 

== square of mean diameter. 

By Rule 6, p. 68, this cask will hold — 

28— 20 = 8 X .65 = 5.2 + 20 = 25.2 X 25.2 X 40 X. 0034 = 86.36 
gallons. 

When the cask is in the form of the middle frustum of a spheroid. 
I D2+ J c?2 = square of mean diameter. 

And a cask of this form, having the same head diameter, bung 
diameter, and length as the preceding, will hold — 

2X282 + 202^ ^Q ^ QQ3^ ^ g^ 216 gaUons. 

o 

When the cask is in the form of the middle frustum of a parabolic 

spindle. 

I D2 + J c?2 — -i-^ (D j^ d)~ =z square of mean diameter. 

And a cask of this form, having the same head diameter, bung 
diameter, and length as the preceding, will hold — 

522i + 133i= 656 — 8.533 = 647.467 X 40 X .0034 = 88.055 gals. 

When the cask is in the form of two equal frustums of a paraboloid, 

JD2 + 4 ^2= square of mean diameter. 

And a cask of this form, having the same head diameter, bung 
diameter, and length as the preceding, will hold — 

282 + 202 

g X 40 X .0034 = 80.51 gallons. 

■ When the cask is in the form of the equal frustums of a cone. 



4 D2 + ^ c?2 — ^ (D vT^ 6?)2 = square of mean diameter. 
Or, ^D2+J^2_|. jD^^ u u u 

Or,DX^ + i(D^^)2= ** '' '' 

And a cask of this form, having the same head diameter, bung 
diameter, and length as the preceding, will hold — 

28 X 20 + 21j X40X.0034 = 79.06 gals. 
6* 



66 GAUGING. 

To find the contents of a cask the same as ivouhl be given by the 
gauging rod. 

The gauging rod is constructed upon the principle that the cube 
of the diagonal of a cask, in inches, multiplied bj ^5^ J^u, equals the 
contents of the cask, in Imperial gallons. 

The contents in wine gallons of either of the aforementioned 
casks, therefore, by the gauging rod, would be — 

31.241' X .0027 = 82j gals. 

The decimal coeflicient to take the place of .0027, for finding tho 
contents of a cask in the form of the middle frustum of a spheroid 
= .00202G ; and for finding the contents of a cask in the form of tGe 
equal frustums of a cone = .002593. And between these extremes 
lies the decimal for other casks, or casks of intervening figures. 

To find the diagonal of a cask^ wlien the interior is inaccessible. 

Rule. — From the bung diameter subtract lialf tho difleronce of 
the bung and head diameters, and to the square of the remainder 
add the square of half the length of tho cask, and the square root 
of the sum will be the diagonal. 

ExAMPLK. — AVhat is the diagonal of a cask whoso bung diameter 
is 28 inches, head diameter 20 inches, and length 40 inches ? 

28 - 20 = 8 -^ 2 = 4 , and 23 — 4 = 24 , t h en 

V (24- + 20') = 31.241 inches. Ans. 

To find the length of a cask^ the head diameter ^ bung diameter and 
diagonal lacing given. 

V ( diagonal^ — D — — ^ — ) = 4 ^• 

And the interior length of a cask, whose /interior head diameter, 
bung diameter and diagonal, are iis the preceding, will be 

V(31.24P-.24-') =20 X 2 = 40 inches. 

To find the solidity of a sphere. 
D- X H D X .7854 = cu])ic contents, D being the diameter. 

To find the solidity of a spherical frustum . 

( h'^-\-d'^\ 

\ 5 ^i" + 2 y X /i X .7854 = cubic contents, b and d Ixjing tho 

bases, and h the height. 

Note. — For Rules in detail pertaining to the foregoing figures, ana lor ciu'.r ni:urce, 
gee Mii:NSU£ATioN ok Solids. 



TILLAGE. (57 



ULLAGE. 



The ullage or ivanlagc of a cask is the quantity the cask lacks of 
•oiug lull. 

To find the ullage of a standing cask, wlicn tlie cask is half full or tnorc. 

Rule. — To the square of the liead diameter, add the square of 
the diameter at the sui'face of the liquor, and the square of twice 
the diameter midway between the surface of the liquor and the upper 
head, and divide the sum hy G ; the quotient, multiplied by the 
distance from the surface of the liquor to the upper head, multiplied 
by .0034, will give the ullage in wine gallons. 

Ex.wfPLK. — The diametei^s are as follows — at the upper head, 20 
inches ; at the surfaoe of the liquor. l!2 inches ; and at a point midway 
between these, 121 1 inches ; and tiie distance from the upper head 
tv> the surface of the liquor is o inches ; required the ullage. 



(i!0- -f 22-'+ 21.25 X 2") -^ = 4iS.37 X ^ X -0034 = 7.62 gal- 
lons. A}u<. 

When the cask is standing, and less than half full, to find tlic ullage. 

KrLK. — Make use of the bung diameter in place of the head 
diameter, and proceed in all respects as directed in the last Rul(\ 
and add the quantity found to half the capacity uC the cask ; the 
sum will l)'e the ullage. 

ExAXiTLE. — The bung diameter is 28 inehos ; the diameter at the 
[surface r»f tiie liquor, below tlie bung, is 26 inches ; the diameter 
midway between the bung and the surface of the liquor is 27.3 
inches ; and the distance from the surface of the liquor to the bung 
diameter is 5 inches ; required the quantity the cask lacks of being 
half full ; and also the ullage of the cask, its capacity being 80. 2d 
siallons. 



(28-' +20)-' + 27.3 X 2') ~ = 740.2 X '> X .0034 = 12.58 gal- 
hmv*^ less than h full. Ans. 
And, 86.21) -f- 2 = 43. lo -|- 12.38 = 55.73 gallons ullage. Ans, 

When the cask is vpon its lilge, and half full or nure, to find the ullage. 

Rule. — Divide the distance from the bung to the surfiice of the 
liquor — (the h.eight of the empty 'segment) — by the whole bung 
iliaiuetor, and take the quotient as the height of the segment of a 
circle whose diameter is 1, and tind the area of the segment; mul- 
tiply the area by the capacity of the cask, in gallons, and that 
product by 1.25 ; the last product will Ih} the ullage, in gallons, as 



68 tTLLAGE. 

found by the aid of the xoantage-rod ; and will be correct for caskB 

of the most comiuon form. 

Note. — The area of the segment of a circle = 

(ch'd i arc -f h ch'd J arc -|- ch'd seg.) X height scg. X xV*! ▼OT nearly. 
And, having the diameter of the circle and the height of the segment giTen, the cboid 
of half the arc, and the chord of the segment may be found, thus — 
radius — heiaht = cosine ; radius" — cosine" =8ine"; ^{sine )^2 = eh^do/i€g. 
sine" 4- height teg.^ = cVd i arc"^ and s/ (ch'd J arc'2) = ch'd \ arc. 

Example. — The bung diameter is 2.^ inches, the height of tho 
empty sogiHont 5.0 inches, and the capacity of the cask 86.26 gal- 
lons ; required the ullage of the cask, in gallons. 

5.6 -^ 28 = .2 = height of seg., diameter as 1. 

1 H- 2 = .5 = radius. 

.5 — .2 = .3 = cosine. 

.5- — .3- = .16 = sine^, or square of half the base of the segment. 

V-l^J = .4 X 2 = .8 = chord of segment, or base of segment. 

.4- -{- -2- = .2 = squiire of chord of half the arc. 

V.2 = .4472 = chord of half tiie arc, then — 

.4472 -T- 3 = .1491, and .1491 + .4472 + .8 X 2 X iS = .1117, 
area of segment, and 

.1117 X 86.26 X 1.25 = 12 gallons. Ans, 

When the cask is upon its bilge, and less than half fully to find the 

ullage. 

Rule. — Divide the depth of the liquor by the bung diameter, and 

proceed in all rcsjx^cts as directed in the last Rule ; tlien subtract 
the quantity found from the capacity of tlie cask, and the diflcrenco 
will bo the ullage of the cask. 

To find the quantity of ii^juor in a cask by its weight, 

ExAxn^LE. — The weight of a cask of proof spirits is 300 lbs., and 
the weight of the empty cask {tare) is 32 lbs. How many gallons 
are there of the liquor ? 

300 — 32 = 268 -h 7,732 = 34? gallons. Ans. 

Ciistomary Rule by Freighiijig Merchants, for finding the cubic meas- 
urement of casks. 

Bung diameter" X t length of cask = cubic measurement. 

Note. — One cubic foot contains 7.4S05 wine gallons. 

' * For several Rules in detail, for finding the area of the segment of a circle, sec Gkom- 
KTRY — Mensuration of Superficies. 



TONNAGE. 69 



TONNAGE. 

GOVERNMENT MEASUREMENT. 



len|];th — - % breadth X breadth X <3epth 
jjr- = tonnage. 

In a double-decked vessel, the length is reckoned from the fore 
part of the main stem to the after side of the stempost above the 
upper deck ; the breadth is taken at the broadest part above the 
main wales, and half this breadth is taken for the depth. 

In a single-decked vessel the length and breadth are taken as for 
a double-decked vessel, and the distance between the ceiling of the 
hold and the under side of the deck plank is taken as the depth. 

Example. — The length of a double-decked vessel is 260 feet, and 
the breadth is GO feot ; required the tonnage. 

260 — -6-\x-^ = 224 X GO X ¥" = 403200 -^ 95 = 4244.2 tons. Ans. 

Example. — The length of a single-decked vessel is 180 feet, the 
breadth 34 feet, and depth 18 feet ; required the tonnage. 

180 — f of 34 = 159.G X 34 X 13 -i- 95 = 1028.16 tons. Ans. 

carpenter's MEASUREMENT. 

For a double-decked — 

length of keel X breadth main beam X 4 breadth 
^ = tonnage. 

For a single-decked — 

length of keel X breadth main beam X depth of hold 

• ^ = tonnage. 



CONDUITS OR PIPES. 



OF CONDUITS OR PIPES. 

Pressure of Water in Vertical Pipes, <^c. 

h = height of column in inches ; o = circumference of column in inches J 
t = thickness of pipe in inches equal in stren«^th to lateral pressure at baso 
of column ; w = weight of a cubic inch of water in pounds ; C = cohesive 
strength in pounds per inch area of transverse section of the material of 
which the pipe is composed — table, p. 72. 

Ao =: area of interior of pipe in inches ; hw=. prcssuro«in pounds per 
square inch at the base of the column, or maximum lateral pressure in 
pounds per square inch on the pipe tending to burst it ; how = maximum 
lateral pressure in pounds on the pipe, tending to burst it at the bottom ; 
and how -^ 2 = mean lateral pressure in pounds on the pipe, or j>re88uro 
in pounds on the pipe tending to burst it at half the height of the column. 

how -^ C =^ i; how-^t^^C; Ct -i- o w = h ; Ct -^ hw = o. 

NoTR. — The rdiable cohesion of a material is not above \ it« ultimale force, as given 
In the Table of Cohesive Forces. By experiment, it ha.s l)een found that a caiji iron pipe 
15 inches in (hameicr and i of an incli thick, will support a hoiiii of water of H'XJ fret ; and 
that one of the same diameter made of oak, ant! two inches thick, will supi^ri a head of 
ISO feet : 12(»U(I Iba. per 8<|u«re inch for ciusi iron, 12(10 for o»k, 750 f<.r Icatl, are cotmled 
safe eatima!n.s. The ultimate cohesion of an alloy, com|K)sed of lead S parts and zinc 1 
part, is 3(KXI pounds per sfpiare inch. 

Concerning the Discharge of Pipes, djc. 

Small pipes, whether vertical, horizontal, or inclined, under equal 
heads, discharp^e proportionally loss water than larpe ones. That 
f«rm of pipe, therefore, which presents the least perimeter to its area, 
other tluniTs being equal, will irive the greatest discharge. A round 
pipe, consequently, will discharge more water in a given lime than a 
l)ipe of any other form, of equal area. 

The greater the length of a pipe discharging vertically, the greater 
the discharge, liecause the friction of the particles against its sides, 
and consequent retardation, is more than overcome by the gravity of 
the fluid. 

The greater the length of a pipe discharging horizontally, the less 
proportionally will be the discharge. The proportion compared with 
a less length is in the inverse ratio of the square root of the two 
lengths, nearly. 

Other things being equal, rectilinear pipes give a greater discharge 
than curvilineai, and curvilinear greater than angular. The head, 
the diameters and the lengths being the same, the time occupied in 
passing an equal quantity of water through a straight pipe is i), 
through one curved to a semicircle 10, and through one having one 
right angle, otherwise straight, 11. All interior inequalities and 
roughness should be avoided. 

It has been ascertained that a velocity of 60 feet a minute (1 foot 
a second) through a horizontal pipe, 4 inches in diameter and 100 feet 



CONDUITS OR PIPES. 71 

in length, is produced by a head 21 inches, only j of an inch above 
the upper surface of the orifice ; and that, to maintain an equal 
velocity through a pipe similarly situated, of equal length, having a 
diameter of | inch only, a head of 1^^ feet is required. To increase 
the velocity through the last mentioned pipe to 2 feet a second, 
requires a head 4|§- feet ; to 3 feet, a head of lOy^^" ' ^^ "^ ^^^^j ^ 
head of 17-ig-, &c. " 

From the foregoing, the following, it is believed, reliable rules, are 
deduced. 

To find the velocity of water passing through a straight horizontal pipe 
of any length and diameter, the head, or height of the fluid above the 
centre of t lie orifice, being known. 

Rule. — Multiply the head, in feet, by. 2500, and divide the product 
by the length of the pipe, in feet, multiplied by 13.9, divided by the 
interior diameter of the pipe in inches ; the square root of the quotient 
will be the velocity in feet per second. 

Example. — The head is 6 feet, the length of the pipe 1340 feet, 
and its diameter 5 inches ; required the velocity of the water passing 
through it. 

2500 X 6 == 15000 -r- (J-ii^ o^-L3--9 ) = ^4.03 — - = 2 feet per 
second. Ans. 

To find the head necessary to produce a required velocity through a pipe 
of given length and diameter. 
Rule. — Multiply the square of the required velocity, in feet, per 
second, by the length of the pipe multiplied by the quotient obtained 
by dividing 13.9 by the diameter of the pipe in inches, and divide the 
product thus obtained by 2500 ; the quotient w411 be the head in feet. 

Example. — The length of a pipe lying horizontal and straight is 
1340 feet, and its diameter is 5 inches ; what head is necessary to 
cause the water to flow through it at the rate of 2 feet a second ? 
2^ X 1340 X -f — -T- 2500 = 6 feet. Ans. 

To find the quantity of water flowing through a pipe of any length and 

diameter. 
Rule. — Multiply the velocity in feet per second by the area of the 
discharging orifice, in feet, and the product is the quantity in cubic 
feet discharged per second. 

Example. — The velocity is 2 feet a second, and the diameter of 
the pipe 5 inches ; w'hat quantity of water is discharged in each 
second of time'? 
5 ~- 12 = .4166, and .4166- X .7854 X 2 = .273 cubic foot. Ans. 



72 MISCELLANEOUS PROBLEMS. 

MISCELLANEOUS PROBLEMS. 

To find the specific gravity of a hodxj heavier than water. 

Rule. — Weigh the body in water and out of water, and divide the 
weight out of water by the difTerence of the two weights. 

Example. — A piece of metal weighs 10 lbs. in atmosphere, and 
but 8i in water ; required its specific gravity. 

10 — 8.25 = 1.75, and 10 -7- 1.75 = 5.714. Ans, 

To find the specific gravity of a body lighter than water. 
Rule. — Weigh the body in air; then connect it with a piece of 
metal whose weight, both in and out of water, is known, and of suf- 
ficient weight that the two will sink in water ; and find their combined 
weight in water ; then divide the weight of the body in air by the 
weight of the two substances in air, less the sum of the dillcrence of 
the weight of the metal in air and water and tlie combined weight of 
the two substances in water, and the quotient will be the specific 
gravity souglit. 

Example. — The combined weight, in water, of a piece of wood, 
and piece of metal, is 1 lbs. ; the wood weighs in atmosphere 10 lbs. ; 
and the metal in atmosphere 12, and in water 11 lbs. ; required the 
specific gravity of the wood. . 

10 -^ (10 + 12 — 12 Lnll + 4) = .588. Ans. 
To find the specific gravity of a fiuid. 
Rule. -^Multiply the known specific gravity of a body by the dif- 
ference of its weight in and out of the fiuid, and divide the product by 
its weight out of the fiuid ; the quotient will be the specific gravity of 
the fluid in which the body is weighed. 

Example. — The specific gravity of a brass ball is 8.6 ; its weight 
in atmosphere is 8 oz., and in a certain fluid 7.1 oz. ; required the 
specific gravity of the fiuid. 
8 — 7.25 = .75, and 8.G X -75 = 6.45, and 6.45 -^ 8 = .806. Ans. 

To find the proportion of one to the other of two simples forming a 
compound, or the extent to which a metal is debased, (the metal ana the 
alloy used being known.) 

The Rule strictly bears upon that of Alligation Alternate, which 
see. 

Example. — The specific gravity of gold is 19.258, and that of 
copper, 8.788 ; an article composed of the two metals, has a specific 
gravity of 18 ; in what proportion are the metals mixed? 
18vr^ 19.258 X 8.788=11.055 
18 v/> 8.788 X li).258 = 177.4, then 



MISCELLANEOUS PROBLEMS. 73 



11.055 -f 177.4 : 11.055 : : 18 = 1.056 copper, >^^^ 
11.055+ 1"7.4 : 177.4 : : 18 = 16.944 gold. S 
Or, 18 — 1.056 = 16.944 gold. Copper to gold as 1 to 16.04 + 

To find the lifting power of a balloon. 
Rule. — Multiply the capacity of the balloon, in feet, by the dif- 
ference of weight between a cubic foot of atmosphere and a cubic foot 
of the gas used to inflate the balloon, and the product is the weight 
the balloon will raise. 

Example. — A balloon, whose diameter is 24 feet, is inflated with 
hydrogen ; w^iat weight will it raise ] 

Specific gravity of air is 1, weight of a cubic foot 527.04 grains; 
specific gravity of hydrogen is .0689. 

527.04 X -0689 = 36.31 grains = weight of 1 cubic foot of hydrogen. 
527.04 — 36.31 = 490.73 grs. = dif. of weight of air and hydrogen, 
243 X -5236 = 7238.24 = capacity in cubic feet of balloon. 
Then, 7238.24 X 490.73 = 3552021 grs. = ^"VWo— = ^^Vo ^^s. 

Ans, 

To find the diameter of a balloon that shall be equal to the raising of a 
giixn weight. 

The weight to be raised is 507y*^ lbs. 

5"07T4~X 7000—490.73 = 7233.24, and 7238.24 -h .5236 = ^ 13824 • 

= 24 f(?et. Ans. 

To find the thickness of a concave or hollow metallic ball or globe, that shall 
have a given buoyancy in a given liquid. 

Example. — A concave globe is to be made of brass, specific grav- 
ity 8.6, and its diameter is to be 12 inches ; what must be its thick- 
ness that it may sink exactly to its centre in pure water? 

Wei ght of a cubic inch of w ater .036169 lb. ; of the brass .3112 lb. 
Then, 12^ X -5236 X .036169 ~- 2 = 16.3625 cubic inches of water 
to be displaced. 

16.3625 -r- .3112 = 52.5787 cubic inches of metal in the ball. 

la^X 3.1416 = 452.39 square inches of surface of the ball. 
And, 52.5787 -r- 452.39 = .1162 -f == ^ inch thick, full. Ans, 

To cut a stjuare sheet of copper, fin, etc., so as to form a vessel of the 
greatest cubical capacity the sheet admits of 

Rule. — From each corner of the sheet, at right angles to the side, 
cut ^ part of the length of the side, and turn up the sides till the 
corners meet. 

7 



74 



COMPARATIVE COHESIVE FORCE. 



Comparative Cohesive Force of Metals, Woods, and other substances. 
Wrought Iron {medium quality) being the unit of comparison, or 1 ; 
the cohesive force of which is 60000 lbs. per inch, transverse area. 



Wrought iron, . 


1.00 


Ash, white, 


.23 


** wire, . 


1.71 


'' red. 


.30 


Copper, cast, . 


.40 


Beech, 


.19 


'' wire, . 


.70 


Birch, 


.25 


Gold, cast, 


.34 


Box, . . . . 


.33 


** wire. 


.51 


Cedar, 


.19 


Iron, cast, (average). 


.38 


Chestnut, sweet. 


.17 


Lead, '* 


.015 


Cypress, . 


.10 


'' milled, . 


.055 


Elm, 


.22 


Platinum, wire, 


.88 


Locust, . 


.34 


Silver, cast, 


.GO 


Mahogany, best. 


.36 


** wire. 


.08 


Maple, 


.18 


Steel, soft. 


2.00 


Oak, Amcr., white, . 


.19 


'' fine, 


2.25 


Pine, pitch. 


.20 


Tin, cast block, 


.083 


Svcamor*', 


.22 


Zinc, '' 


.043 Walnut, . 


.30 


'' sheet. 


.27 1 Willow, . 


.22 


Brass, cast. 


.75, Ivory, 


.27 


Gun metal. 


.50 i Whalebone, 


.13 


Gold 5, copper 1, 


.83 1 Marble, . 


.15 


Silver 5, '^ 1, 


.80 Glass, plate, . 


.10 


Brick, . 


.05 1 Hemp (i!)rrs. glued, . 


. 1.53 


Slate, 


.20 







The strnii^th of white oak to cast iron, is as 2 to 9. 
The s/tjrncss of ** " *' " is as 1 to 13. 

To determine the tccight, or force, in pounds, necessary to tear asun- 
der a bar, rod, or piece of am/ oT the nbiwc iinmrd su/>sfn7iet s, of any 



Rule. — Multiply the cuniiiarativc cohesive force of the substance, 
as given in tlie table, by the cobcsive force per s<]uai;e inch, area of 
cross section (60000 lbs.) of wrought iron, which gives the cobcsive 
force of 1 square inch area of cross section of the substance whoso 
power is sought to be ascertained, and the product of 1 square inch 
thus found, multiplied by area of cross section, in inches, of the rod, 
piece, or bar itself, gives the cohesive force thereof. 

Alloys having a tenacity greater than the sum of their constifurnts. 
Swedish copper 6 pts., Malacca tin 1 ; tenacity per sq. inch, 64000 lbs. 
Chili copper 6 pts., Malacca tin 1 ; *« ' u a fjooOO " 

Japan copper 5 pts., Banca tin 1 ; *' " " 57000 ** 

Anglesea copper 6 pts., Cornish tin 1 ; ** '* *' 41000 '' 



LINEAR DILATION OF SOLIDS BY HEAT. 75 

Common block-tin 4 pts., lead 1, zinc 1 ; tenacity pert sq. in., 13000 lbs. 
Malacca tin 4 pts.,regulus of antimony 1; ^* '' '' 12000 " 

Block-tin 3 pts., lead 1 part ; " '' '' 10000 '' 

Block-tin 8 pts., zinc 1 part ; '' '' " 10200 '' 

Zinc 1 part, lead 1 part; '' '' " 4500 " 

Alloys having a density greater than the mean of their constituents. 

Gold with antimony, bismuth^ cobalt, tin, or zinc. 

Silver with antimony, bismuth, lead, tin, or zinc. 

Copper with bismuth, palladium, tin, or zinc. 

Lead with antimony. 

Platinum with molybdinum. » 

Palladium with bismuth. 

Alloys having a density less than the mean of their constituents. 
Gold with copper, iron, iridium, lead, nickel, or silver. 
Silver with copper or lead. 
Iron with antimony, bismuth, or lead. 
Tin with antimony, lead, or palladium. 
Nickel with arsenic. 
Zinc with antimony. 

RELATIVE POWER OF 



188 

158 

155 

83 



LINEAR DILATION OF SOLIDS BY HEAT. 

Length which a bar heated to 212° has greater than when at the tan- 
peraturc of 32°. 

Iron, wrought, . 
Lead, 





1 rii\^ 1 1 1 ^ 

{the mass of each being 


equal.) 


Copper, . 


, , 


. 1000 


Platinum, 


Gold, . 




. 93f> 


Iron, 




Silver, 


. 


. 736 


Tin, 




Zinc, 




. 285 


Lead, 





Brass, cast, . . .0018671 

Copper, . . . .0017674 

Fire brick, . . .0004928 

Glass, . . . .0008545 

Gold, . . . .0014880 

Granite, . . . .0007894 

Iron, cast, . . .0011111 



Marble, 

Platinum, 

Silver, 

Steel, 

Zinc, 



.0012575 

.0028568 
.0011016 
.0009342 
.0020205 
.0011898 
.0029420 



Note. — To find the surface dilation of any particular article, double its linear dilation, 
and to find the di ation in volume, triple it. To find the elongation in linear inches per 
linear foot, of anj particular article, multiply its respective linear dilation, as given in the 
lABLB. by 12. 



76 



EFFECTS OF IIEAT. 



MELTING POINT OF METALS AND OTHER BODIES. 

Linie, 'palladium, platinum, porcelain, rhodium, silex, may be melted 
by means of strono^ lenses, or by tbe hydro-oxygen blowpipe. Co- 
balt, manganese, plaster of Paris, pottery^ iron, nickel, &c., at from 
2700^ to 3250-* Fahrenheit; others as follows : — 





I)c^ee« Fah. 








I>eyT««« f*h. 


Antimony, 




. 809 


Nitre, .... 600 


Beeswax, bleached, . 


. 155 


Silver, 




. 1873 


Bismuth, . 


, 


. 500 


Solder, common, 




. 475 


Brass, 




. 1900 


** plumbers'. 




. 360 


Copper, 




. 1<M)0 


Susrar, 




. 400 


Glass, Hint, 




. 1178 


Sul )hur. 








. 220 


Gold, 




. 2010 


Tal ow, 








. 127 


Lead, 




. 012 


Tin. 








. 442 


Mercury, . 




. —39 


Zinc 








. 680 


Cast iron thorouj^hly melts at 






27hO 


Greatest heat 


of a smith's 


'orgc, (com.) . 






2316 


Weldinc? heat 


of iron. 








1892 


Iron red hot in twilifjht, 


. 






884 


Lead 1, tin 1, 


bismuth 4, melts at . 






. 201 


Lead 2, tii 3, 


bismuth 5, 


it ii 






. 212 


RELATIVE POWER OF DIFFERENT BODIES TO RADIATE HEAT. 


Water, . 




100 


I^ad, bright, . 1 ' 


Copper, . 




12 


Mercury, 


20 


Glass, . 




90 


Paper, white, . 


100 


Ice, . 




85 


Silver, .... 


12 


India ink. 




88 


Tin, blackened, 


100 


Iron, ])olished 


> 


i:> 


" clean. 


12 


Lampblack, 




i()() 


'' scrap! 


>d, . 






10 



Note. — The ix)wcr of a l)otly to rxjlcct heat is inverse to its power of rsuliation. 

BOILING POINT OF LIQUIDS. 
Barometer at 30 in. 
Acid, nitric, 

** sulphuric. 
Alcohol, anhyd., . 

'* 90 per cent.. 
Ether, sulph.. 
Mercury, 

Note. — Barometer at 31 inches, water boils at2130.57; at 29, it boils at21(P.39; at 
2S. il boils at 20So.6'J ; at 27, it boils at 200^ So, and in vacuo it boils at 880. No'liqniil, 
under pressure of the almosphere alone, can \ye heated alxn-e its boiling poinl. At thai 
point tlie steam emitted sustains the weight of the atmosphere. 



253<^ Oils, essential. 


avp.. 


318^ 


600^ '' turpentine, . 


316^ 


1G8.5^ '' linseed. 




Oin^ 


174^ Phosphorus, 


. 


551- 


97^1 Sulphur, 




500^ 


656^ [Water, 


. 


212^ 



EFFECTS OF HEAT, ETC. 



77 



FREEZING POINT 

Acid, nitric, . . — 55^ 

** sulphuric, . P 

Etiier, .... — 47- 

Mercury, . . . — 39- 

Milk, .... 30- 

Oil, cinnamon, . . 30" 

** fennel, ... 14^ 

** olive, . . . 30-" 

Note. — Water expands in freezing .11, or -^ its bulk. 



OF LIQUIDS. 

Oil, linseed, avg.. 
Proof spirits. 
Spirits turpentine, 
A^inegar, 
Water, 
Wine, strong, 
Rapesecd Oil, 



—IF 

32^ 

20^ 
25^ 



EXPANSION OF FLUIDS BY BEING HEATED FROlM 32'^ TO 212^ 

Atmospheric air, -^^^ per each degree, = .375 

Gases, all kinds, -^^-^ '' •' *' 

Mercury, exposed, . . . . . . .018 

Muriatic acid, (sp. gr. 1.137,) . . . .060 

Nitric acid, (sp. gr. 1.40,) 110 

Sulphuric acid, (sp. gr. 1.85,) .... .060 

"• ether, — to its boiling point, . . .070 

Alcohol, (90 per cent.,) <' ^^ . . .no 

Oils, fixed, 080 

" turpentine, ...... .070 

Water, 046 



RELATIVE POWER OF SUBSTANCES TO CONDUCT HEAT. 



Gold, 
Silver, . 
Copper, . 
Platinum, 
Iron, 



1000 
973 
898 
381 
374 



Zinc, 
Tin, 
Lead, 
Porcelain, 
Fire brick, 



363 

304 

180 

12 

11 



Note. — Diiferent woods have a conducting power in ratio to each other, as is Iheir 
respective specific gravities, the more dense having the greater. 



METALS IN ORDER OF DUCTILITY AND MALLEABILITY. 



Ductility, 

1. Platinum. 

2. Gold. 

3. Silver. 

4. Iron. 

5. Copper. 

6. Zinc. 

7. Tin. 

8. Lead. 



Malleability. 

1. Gold. 

2. Silver. 

3. Copper. 

4. Tin. 

5. Platinum. 

6. Lead. 

7. Zinc. 

8. Iron. 



7* 



78 



NUTRITIVE AND ALCOHOLIC PROPERTIES OF BODIES. 



Quantity per cent, by weight of Nutritiovs Matter contained in different 
articles of Food. 



Articles. 






per cl. 


Articles. 


per ri. 


Lentils, .... 94 


Oats, 


74 


Peas, 






93 


Meats, avg., 


35 


Beans, 






92 


Potatoes, . 


25 


Corn, (maize,) 






89 


Beets, 


14 


Wheat, . 






85 


Carrots, . 


10 


Barley, . 






83 


Cabbage, . 


7 


Rice, 






88 


Greens, . 





Rye, 






79 


Turnips, white. 


4 



Specific gravity, and quantity per cent., by volume, of Absolute Alcohol 
contomcd, necessary to constitute tlie followijig named unadulterated 
articles. 

.„■ I . 8p. ^T«v. Per cert. 

Absolute Alcohol, (anhydrous,) . .7939 100 

Alcohol, hirjhcst by distillation, .825 92.6 

** commercial standard, . .S335 90 

Proof Spirits, — standard, . . . .9254 54 

Quantity per cent., by volume, (general average) of Absolute Alcohol 
contained in different pure or unadulterated laquorSf Wines , ^c. 



Liqiion, &c. 


per cent. 


Wines. 






per cenl. 


Rum, 


50 


Port, . . • 22 


Brandy, . 


50 


Madeira, 




20 


Gin, Holland, . 


48 Sherry, 




18 


Whiskey, Scotch, 


50 Lisbon, . 






17 


Irish, 


50 


Claret, . 






10 


Cider, whole. 


9 


Malaga, . 






16 


Ale, 


8 


Champagne, 






14 


Porter, 


7 


Burgundy, 






12 


Brown Stout, . 


6 


Muscat, 






17 


Perry, . . 


9 


Currant, 






19 



Proof of Spirituous Liquors. 

The weight, in air, of a cubic inch of Proof Spirits, at 60° F., is 
233 grains ; therefore, an inch cube of any heavy body, at that tempera- 
ture, weighing 233 grains less in spirits than in air, shows the spirits 
in which it is weighed to be proof If the body lose less of its weight, 
the spirit is above proof, — if more, it is below. 



COMPAEATIVE WEIGHT OF TDIBEE. 



79 



Comparative Weight of different kinds of Timber in a green and per- 
fcctly seasoned state. 
Assuming the weight of each kind destitute of water to be 100, that 
of the same kind green is as follows : — 



Ash, 


153 


Cedar, . 


148 Maple, red, . 


149 


Beech, . 


174 


Elm, swamp, . 


198 


Oak, Am., 


151 


Birch, . 


169 


Fir, Amer., . 


171 


Pine, white, . 


152 



Note. — Woods which have been felled, cleft and housed for 12 monlhs, still retain 
from 20 to 25 per cent, of water. They therefore contain but from 75 to 8<J p)er cent, of 
heating mailer : and it will require from 23 to 2ii per cent, the weight of such woods to 
dispel the water they contain. They are, therefore, less valuable by weight, as fuel, by 
this per cent., than woods perfectly free from moi.sture. They never, however, contain, 
exposed to an ordinary atmosphere, less than 10 per cent, of water, however long kept; 
and even though rendered anhydrous by a strong heat, Ihey again imbibe, on exposure to 
the atmosphere, from 10 to 12 per cent, of dampness. 

Relative 'power of different seasoned Woods, Coals, djfc., as fuel, to pro- 
duce heat, — the Woods supposed to he seasoned to mean dryness, 
(77J per cent.,) and the other articles to contain but their usual quan- 
tity of moisture. 





Ratio of Heatin? 
Power per eqiiaf 




Bulk. 


Wei'hU 


Hickory, shell-bark, 


1.00 


1.00 


*^ red-heart, 












.81 


.99 


Walnut, com. 












.95 


.98 


Beech, red, 














.74 


.99 


Chestnut, 














.49 


.98 


Elm, white, 














.58 


.98 


Maple, hard. 














.66 


.98 


Oak, white. 














.81 


.99 


'; red, . 














.69 


.99 


Pine, white. 














.42 


1.01 


*' yellow. 














.48 


1.03 


Birch, black. 














.63 


.99 


" white, 














.48 


.99 


Coal, Cumberland, (bit.) 












2.56 


2.28 


*' Lackawanna (anth.) 












2.28 


2.22 


" Lehigh, 












2.39 


2.03 


'' Newcastle, (bit.) 












2.10 


1.96 


" Pictou, (bit.) 












2.21 


1.91 


'' Pittsburgh, (bit.) 












1.78 


1.82 


" Peach IMountain, (anth. 


) ' 










2.69 


2.29 


Charcoal, .... 












1.14 


2.53 


Coke, Virginia, natural, 












1.89 


2.12 


'* Cumberland, 












1.31 


2.25 


Peat, ordinary, . 














.62 


Alcohol, common, 














2.02 


Beeswax, yellow, 












2.90 


Tallow, , 


• 












» 


3.10 



80 



ILLUMINATION. 



Note. — By help of the preceding table, the price of either one article beinir Known, 
the relative or par value of either other, as fuel, may be reailily ascertained : — Exampli 5 

Maple (6G) : $5.00 : : Pine (12) : $3.18. 



ILLUMINATION — ARTIFICIAL. 

The following Table shows : — 

1. The materials and methods of usin^ — column Matfriah. 

2. The comparative maximum intensity of liglit afforded by each 
material, used or consumed as indicated, — column Intnmtics. 

3. The \vein[^ht, in grains, of material consumed per hour, by each 
method respectively, in producinfj its respective light, or light of in- 
tensity ascribed — column Wciixht. 

4. Tlie ratio of weight required of each material, under each spe- 
cial metbod of consumption, for the productioii of equal lights in 
equal times — column Ratios. 

Materials. Intena. Wriirhl. Ratiuk. 

Camphene* Paragon Lamp, ... 16. 853 L 

Sperm Oil, Parker's beating Lamp, 11. 09(1 1.19 

** *» Mech. or Carcel ** 10. 815 1.53 

'' '' French annular ** . . 5. 513 2.04 

'' '' Common band ** . . 1. 112 2.10 

W/ialc '' p'fd., P's heating '^ . . 1). 780 1.63 

Wax Candles, 3's or 4*8, 15 in. or 12 in., . 1. 125 2.3.5 

** ** G's, 9 in., .... .92 122 2.50 

Sper77i '' 4's, 13.i in., .... 1. 142 2.06 

Strarinc'' 4\s, 13.^ in., .... 1. 168 3.15 

Tallow,'' dipped, lO's, . ... .70 150 4.02 

** ** mould, lO's, .... .66 132 3.75 

8's, .... .57 132 4.35 

6's, ... .79 163 3.87 

4*s, 13J in., . . I. 186 3.49 

*' Coal Gas,^^ intensity being ... 1. 740 

Note. — The consuinpiion of 1. 13 cubic feet of gas per hour, cives a light equal to ono 
wax candle,— the cousJumpliou of 1.9G cubic feet j^er h(uir, a light espial lo fi»ur wax can- 
dles, and the consumption of ^ cubic feet per hour, a light aiual lo ten wax candles. A 
cubic foot of giia weighs 518 irrains. 

The average yield of bi carbureted hydrogen — defiant gaa — Coal Gaa, obtained from 
the following articles, is as annexed. 

1 lb. Bituminous Coal, 1J cubic feel. 

1 lb. Oil, or Oleine, 15 " •* 

1 11). Tar, 12 ** " 

1 lb. Rosin, or Pitch, 10 " " 

A pipe whose interior diameter is i inch, will supply gas equal in illuminating power to 
20 wax randies. 



* 1 lb. Camphene, 
1 lb. Sperm Oil, . . 
1 lb. Whale " pTd., 



=- 1 



^l^pnn. 

1 " 

1 (I 
2^ 



ILLTTMmATION. 81 

• 

By the foregoing table, it is readily seen in what ratio tlie several 
intensities, furnished hy the different methods, stand one to another, 
— that the French annular lamp, for instance, has a maximum povver 
= half tiiat of the mechanical, or -^^ that of the camphene, or 5 
wax candles, 3 to the lb., — that the camphene, at its maximum 
power, yields an intensity equal to that afforded by 16 -h -57, = 28 
tallow candles, moulds, 8 to the lb., — that as the intensity of a six 
wax candle, 13 in., is .92, and that of an eight mould tallow .57, 57 
candles of the former yield an intensity equal to that aflorded by 92 
of the latter, &c., &:c. 

The quantity of material consumed in any given time by either 
of the foregoing methods, in the production of any given intensity of 
light, is readily ascertained by lielp of the preceding table. Suppose, 
for example, an intensity equal to that afforded by 1 camphene para- 
gon lamp at its greatest power, is required, and for three hours, and 
that it is proposed to produce the same by tallow candles, moulds, 10 
to the lb. ; the quantity by weight of candles consumed in the produc- 
tion is required, and, consequently, the number of lights that must 
be used. 

illustration. 

Intens. of camph., (16) -^ intensity of candles, (.60; =2-1 candles, and era. in 1 h. by 
1 candle, (132) X "^i X -^ l^ours = I lb. 5-iJ oz. Ans. 

The economy of use, as between any two materials, under either 
their respective forms, or methods of consumption, for the production 
of equal lights in equal times, and therefore for the production of any 
intensity, is also, by help of the given table, easily learned, the 
market price of both being known ; and, thereby, the per ccnt.^ if any 
difference exist, in favor of the more economical, or less expensive 
of the two, Tnay be found. To illustrate : — 

1. The price of camphene is 10 cents a pound, and that of sperm 
oil, 15 ; the economy of use as between the two for the production of 
equal lights — equal intensities in equal times, greater or less — the 
former consumed in the paragon lamp, and the latter in Parker's heat- 
ing, is desired, and ih^ per cent, in favor of the less expensive. 

10 X 1 = 10, and 15 X 1.19= 17.85; showing the economy to 
be in favor of the camphene — showing it so to an extent 17.85 — 10 
= TrVb"' ^^ ^^ ^^ extent 7 cts. 8 J mills per 17 cts. 8|- mills — to 
an extent, therefore, 

17.85 : 7.85 : : lOO = 44 per cent. Ans. 

2. The price of sperm candles, 4's, is 40 cts. a pound, and the price 
of tallow, mould lO's, is 11 cts. It is desired to know which of the 
two, for the production of an intensity nearest obtainable to that af- 
forded by one of the w^ax, but not less than that of 1 wax, is the less 
expensive, and to what extent joer cent. 

By casting the eye to the table, it is readily seen that two tallow 
candles must be employed, the comparative intensity of which 



82 THERMOMETERS. 

is .66 each ; therefore, .66 X 2 = 1.32, and 3.75 X 1-32 = 4.95, 

equivalent weight; consequently — 

40 X 2.66 = 106.4, and 11 X 4.95 = 54.45, and 106.4 — 54.45 « 

51.95. Therefore, 

106.4 : 51.95 : : lOO =48^®^ per cent. Ans. 

Showing that an intensity nearly J greater is afforded by the tallow 
than the wax, and at an expense 49 per cent. less. The same rule 
of practice is applicable, as between any two methods, for equal or 
greater or less intensities, as desired. 



Fahrenheit's, . 

Reaumur's, 

Centigrade, 



THERMOMETERS. 

Boilmo' point. 

212^ 

bO^ 
100^ 



Frtft'ing point. 

32^^^ 
0° 
0^ 



To reduce Reaumur to Fahrenheit. 
When it is desired to reduce the -|- °, (degrees above the zero) : — 

Rule. — Multiply the degrees Reaumur, by 2.25, and add 32° to 
the product ; the sum will be the degrees Fahrenheit. 

When it is desired to reduce the — °, (degrees below the zero) : — 

Rule. — Multiply the — ^ Reaumur by 2.25, and subtract the 
product from 32' ; the diirorencc will be the degrees Fahrenheit. 

Example. — The degrees R. are 40 ; required the equivalent 
degrees F. "* 

40 X 2.25 = 90 -j- 32 = 122°. Ans, 

Example. — The degrees below 0, R., are 10; what are the cor- 
responding degrees F. ? 

10 X 2.25 = 22.5, and 32 — 22.5 = OJ^ Ans. 

Example. — The degrees below 0, R., are 16 ; what point on the 
scale F. corresponds thereto ? 

16 X 2.25 = 30, and 32 — 36 = — 4 ; 4° below 0. Ans. 

To reduce the Centigrade to Fahrenheit. 
Rule. — Multiply the degrees C. by 1.8, and in all other respects 
proceed as directed for Reaumur, above. 

Note. — The zero of Wcilgewood's pyrometer is fixed at the lenipcrature of iron red-hot 
in daylight. = 1077O p., and each degree W. equals 130O F. The instrument is not coq- 
eidered reliable, and is but little used 



HORSE POWER — ANIMAL POWER STEAM. 83 

HORSE POWER. 

A HORSE-POWER, in machinery, as a measure offeree, is estimated 
equal to the raising of 33000 lbs. over a single pulley one foot a 
minute, = 550 lbs. raised one foot a second, = 1000 lbs. raised 33 feet 
a minute. 



ANIMAL POWER. 

A man of ordinary strength is supposed capable of exerting a force 
of 30 lbs. for 10 hours in a day, at a velocity of 24 feet a second, = 
75. lbs. raised 1 foot a second. 

The ordinary working power of a horse is calculated at 750 lbs. for 
8 hours in a day, at a velocity of 2 feet a second, = 375 lbs. raised 
1 foot a second, = 5 times the effective power of a man during asso- 
ciated labor, and 4 times his power per day ; and as machinery may 
be supposed to work continually, = a trifle less than 23 per cent, per 
day of a machine horse-power. 



STEAM. 

Table exhibiting the expansive force and various conditions of steam 
under different degrees of temperature. 



De^ees of 
heal. 


Pressure in 
atmospheres. 


Density. 
Wraeras 1. 


Volume. 
Water as 1. 


Spec, gravity. 
Air as 1. 


Weight of a 

cubic foot in 

grains. 


212 


1 


.00059 


.1694 


.484 


254 


250.5 


2 


.00110 


909 


.915 


483 


270 


3 


.00160 


625 


1.330 


700 


293.8 


4 


.00210 


476 


1.728 


910 


308 


5 


.00258 


387 


2.120 


1110 


359 


10 


.00492 


203 


3.970 


2100 


418.5 


20 


.00973 


106 


7.440 


3940 



[An atmosphere is H-^^ lbs. to the square inch.] 

Note. — By the above table it is seen that any given quantity of steam having a tem- 
perature of 212^ F., occupies a space, under the ordinary pressure of the atmosphere, 
1694 times greater than it occupied when as water in a natural state. It exerts a mechan- 
ical force, consequently, := 1694 times the weight or force of the atmosphere resting on 
the bulk from which it was generated, or resting on l-1694th of the space it occupies. 
A force, if we consider the volume as so many cubic inches, equal to the raising of 2087 
lbs. 12 inches high, by a quantity of steam less than a cubic foot, heated only to the tem- 
peratm-e of boiling water, and weighing but 248 grains, and that, too, the product of a 
single cubic inch of water. 

The mean pressure of the atmosphere at the earth's surface is equal 
to the weight of a column of mercury 29.9 inches in height, or to a 
column of water 33.87 feet in height, = 2116.8 lbs. per square foot, or 



84 



VELOCITY AND FORCE OF WIND. 



14.7 lbs. per square inch. It^ density above the earth is uniformly 
less as its altitude is greater, and its extent is not above 50 miles — 
its mean altitude is about 45 miles; at 44 miles it ceases to rellect 
light. Were it of uniform density throughout, and of that at the sur- 
face, its altitude would be but 5^ miles. Its weight is to pure water 
of equal temi)erature and volume, as 1 to 829. It revolves with the 
earth, and its average humidity, at 40- of latitude, is 4 grains per cubic 
foot. Its weight at GO^, b. 30, compared with an equal bulk of pure 
water at 40', b. 30, is as 1 to 830.1. 



VELOCITY AND FORCE OF WIND. 





Mean TC 


luciiy m 






Mil*** per 


Feel per 


Force in . ytr 


Apprllntiona. 


h..ur: 


•C('UII<i. 


•<juar. : 


Just perceptible. 


2i 


3i 


.032 


Gentle, pleasant wind, . 


44 


fi3 


.101 


Pleasant, brisk gale, 


m 


18J 


.80 


A^cry brisk, ** 


22i 


33 


2.52 


High wind, 


3t>i 


■171 


5.23 


Very high wind, . 


424 


02 i 


8.92 


Storm, or tempest. 


50 


T3J 


12.30 


Great storm, 


no 


86 


17.71 


Hurricane, 


80 


117i 


31.49 


Tornado, moving buildings, &c., 


100 


110.7 


49.20 



The curvature of the earth i3dD.99 inches (.5825 foot) in a single 

statute mile, or 8.05 inches in a geouraphical mile, and is as the 

square of the distance for any distance greater or less, or space 

between two levels ; thus, for three statute miles it is 

1 : 32 : : 6.99 : 51 feet, nearly. 

The horizontal refraction is -j-Vj-. 

Degrees of longit\ide arc to each other in length, as the cosines of 
their latitudes. At the equator a degree of longitude is 00 geographical 
miles in length, at 90^ of latitude it is ; consequently, a degree of 
longitude at 

5^ . . =59.77 miles. 40^ . . = 15.96 miles. 

10^ . . =59.09 *' 50'' . . =38.57 *' 

SO'' . . =50.38 '' 70^ . . ='20. ,52 " 

30^ . . =51.96 '* 85^ . . = 5.23 " 

Time is to longitude 4 minutes to a degree, — ftister, east of any 
given point ; slower, west. 

The mean velocity of sound at the temperature of 33^ is 1100 feet 
a second. Its velocity is increased 4 a foot a second for every degree 



GRAVITATION. 85 

above 33°, and decreased ^ a foot a second for every degree below 
83 . 

In water, sound passes at the rate of 4,708 feet a second. 

Light travels at the rate of 192,000 miles per second. 



GRAVITATION. 

Gravity, or Gravitation, is a property of all bodies, by which 
they mutually attract each other proportionally to their masses, 
and inversely as the square of the distance of their centres apart. 
Practically, therefore, with reference to our Earth and the bodies 
upon or near its surface, gravity is a constant force centred at the 
Earth's centre, and is there continually operating to draw all bodies 
with a uniformly accelerating velocity to that point, and through 
very nearly equal spaces, in equal intervals of time from rest, at all 
localities. 

Putting R^ to represent the Equatorial radius of the earth, and r 
to represent the Polar, and making IV = 39G2.5 statute miles, and 
rzzi 3949.5, which is nearly in accordance with the mean of the 
most reliable measurements of the arcs of a degree of latitude at 
different localities, we have e^= (72'-— r) -^i2'- = .006550751, the 
square of the ellipticity of the earth, and R =: 2R' -^ (2 -|- e^ sinHjj 
the radius at any given latitude /. 

And since the initial velocity due to gravity at the level of the 
sea at the Equator is G = 32.0741 feet per second, or, in other 
words, since a body falling in vacuo at the equator, at the level of 
the sea, describes a space of 1G.03705 feet in the first second of 
time from rest, we have fjr=i\_R' yG) ~ /2]-, the initial velocity at 
the level of the sea at any given radius R ; or g = (22441.2 ~ Ry. 

And finally </= (— ^^)' X (l — Jl^) at any given ra- 
dius it, at any given altitude, 7i, in feet, above the level of the sea. 
Note. — When I, reckoned from the equator, is liigher than 45'*, sin- Z = 

The momentum, or force, with which a falling body strikes, is the 
product of its weight and velocity (the weight multiplied by the 
square root of the product of the space fallen through and 64.33, 
or 4 times 1(^y2) ' thus, 100 lbs., falling 50 feet, will strike with a 
force, 

50 X 04.333=^3216.66 — 56.71 X 100 = 5671 lbs. 

An entire revolution of the earth, from west to east, is performed 
in 23 hours, 56 minutes, and 4 seconds. A solar year = 365 days, 
5 hours, 48 minutes, 57 seconds. 

The area of the earth is nearly 1 9 7,000,000 square miles. Its cmst 

is supposed to be about 30 miles in thickness, and its mean density 5 

times that of water. About f of its area, or 150,000,000 square 

miles, is covered by water. The portions of land in the several 

8 



86 



CHEMICAL ELEMENTS. 



divisions, in square miles, are, in round numbers, as follows, 
viz : — 



Europe, . . 3,700,000 

Australia, . . 3,000,000 



Asia, . . 16,300,000 

Africa, . . 11,000,000 

America, . . 11,500,000 

America is 9000 miles long, or -f^^(j the circumference of the 
earth. 

The population of the globe is about 1,000,000,000, of which there 
are, in 

Asia, . . 456,000,000 I Africa, . . 62,000,000 

Europe, . ii58,000,000 | America, . . 55,000,000 



CHEMICAL ELEMENTS. 

The chemical elements — simple substances in nature — as far as 
has been determined, are 58 in number: 13 non-metallic and 45 
metallic. 

Of the non-metallic, 5 — bromine, chlorine^ fluorine^ iodine, Sind oxy- 
gen, (formerly termed ^^ su])])ur/crs of combustion,^') have an intense 
aHinity for all the others, which they penetrate, corrode, and appar- 
ently consume, always with the i)roduclion, to some extent, of lipht 
and heat. They are all non-conductors of electricity and negative 
electrics. 

The remaining 8 — hydrogen, nitrogen or azote, carbon, boron, sili- 
con, phosphorus, selenium, and sulphur, are eminently susceptible of 
the impressions of the preceding five ; when acted upon by either of 
them to a certain extent, light and heat are manifestly evolved, and 
they are thereby converted into incombustible compounds. 

Of the metals, 7 — potassiiun, sodium, calcium, bar y/ium, lithium, 
strontium, and magnesia, by the action of oxygen, are converted into 
bodies possessed o\' alkali fie properties. 

Seven of them — glucinum, erbium, terbium, yttrium, allumium, zir- 
conium, and thorium, — by the action of oxygen, are converted into 
the earths proper. 

In short, all the metals arc acted upon by oxygen, as also by most 
or all of the non-metallic family. The compounds thus formed are 
alkaline, saline, or acidulous, or an alkali, a salt, or an acid, according 
to the nature of the materials and the extent of combination. 

Metals combine with each other, forming alloys. If one of the 
metals in combination is mercury, the compound is called din amalgam, 

Silicoji is the base of the mineral world, and carbon of the organ- 
ized. 

For a very general list of the metals, see Table of Specific Grav- 
ities. 



CONSTITUENTS OF BODIES. 



87 



TABLE 

Exhibiting the Elementary Constituents and per cent, hy weight of each^ 
in 100 parts of different compounds. 



Compounds. 1 


Constituents 


and per cent. | 


Atmospheric air, . . . c 


HyfJrog-cn. 


Oxyg-en. 


Azote. 


Carbon. 




20.8 


79.2 




Water, pure, 


11.1 


88.9 






Alcohol, anhydrous, 


12.9 


34.44 




52.66 


Olive oil, . 


13.4 


9.4 




77.2 


Sperm'* .... 


10.97 


10.13 




78,9 


Castor ''.... 


10.3 


15.7 




74.00 


Stearine, (solid of fats,) 


11.23 


6,3 


0.30 


82.17 


Oleine, (liquid of fats,) 


11.54 


12.07 


0.35 


76.03 


Linseed oil, . 


11.35 


12.64 




76.01 


Oil of turpentine. 


11.74 


3,66 




84.6 


^' Gzmp/tene," (pure spts. turp.) 


11.5 






88.5 


Caoutchouc, (gum elastic,) . 


10. 






90. 


Camphor, .... 


11.14 


11.48 




77.38 


Oopal, resin. 


9. 


11.1 




79.9 


Guaiac, resin, 


7.05 


25.07 




67.88 


Wax, yellow. 


11.37 


7.94 




80.69 


Coals, cannel. 


3.93 


21.05 


2.80 


72.22 


'' Cumberland, 


3.02 


14.42 


2.56 


80. 


*'- Anthracite, . . h 








93. 


Charcoal, .... 








97. 


Diamond, .... 








100. 


Oak wood, dry, . . . c 


5.69 


41,78 




52.53 


Beech "*'... 


5.82 


42.73 




51.45 


Acetic acid, dry, 


5.82 


46.64 




47.54 


Citric ^' crystals, . 


4.5 


59.7 




35.8 


Oxalic '^ dry, 




79.67 




20.33 


Malic, '* crystals^ . 


3.51 


55.02 




41.47 


Tartaric " dry, 


3. 


60.2 




36.80 


Formic '* ^' 


2.68 


64.78 




32.54 


Tannin, tannic acid, solid, 


4.20 


44.24 




51.56 


Nitric acid, dry, . 




73.85 


26.15 




Nitrous '* anhydrous, liquid, 




61 32 


30.68 




Ammoniacal gas, 


17.47 




82.53 




Carbonic acid *' . 




72,32 




27.68 


Carb. hydrogen gas. 


24.51 






75.49 


Bi-carb. hyd., olefient gas, . 


14.05 






85.95 


Cyanogen , '* 






53.8 


46.2 


Nitric oxyde " 




53. 


47.00 




Nitrous " *< 




36.36 


63.64 




Ether, sulpliuric, . 


13.85 


21.24 




65.05 


Creosote, .... 


7.8 


16. 




76.2 



88 



CONSTITTTENTS OF BODIES. 





Coiutituent* 


and per cent. i 


Compound!. 

Morphia, .... 


Hydroeen.l Uxyrfn. 


A.'oie. 


CnrSon, 


0.37 


16.29 


5. 


72.34 


Quina, — quinine, 


7.52 


8.61 


8.11 


75.76 


Yeratrine, .... 


8.55 


19.61 


5.05 


66.79 


Indigo, .... 


4.38 


11.25 


10. 


71.37 


Silk, pure white, . 


3.91 


34.04 


11.33 


50.69 


Starch, — farina, dextrine, . 


6.8 


49.7 




43.5 


Sugar, .... 


6.29 


50.33 




43.38 


Gluten, .... 


7.8 


22. 


14.5 


55.7 


Wheat, . . . . c 


6. 


44.4 


2.4 


47.02 


Kye, 


5.7 


45.3 


1.7 


47.03 


Oats, 


6.6 


38.2 


2.3 


52.9 


Potatoes, .... 


6.1 


46.4 


1.06 


45.9 


Peas, 


6.4 


41.3 


4.3 


48. 


Beet root, .... 


6.2 


46.3 


1.8 


45.7 


Turnips, .... 


6. 


45.9 


1.8 


46.3 


Fibrin, . . . . d 


7.03 


20.30 


19.31 


53.30 


Gelatin, . . . , d 


7.91 


' 27.21 


17. 


47.88 


Albumen, . . . . <f 


7.54 


, 23.88 


f 15.70 


52.88 



Muriatic acid gas, — Hydrogen 5.53 -{- 94.47 chlorine. 
Sulphuric acid, dry, — Oxygen 79.67 -f- 20.33 sulphur. 
Silicic acid — Silica, dry, — Oxyijcn 51.96 -f- 48.04 silicon. 
Boracic acid — Borax, dry,— " 68.81 -|- 31.19 boron. 

a. The atmosphere, in addition to its constituents as given in the 
table, contains, l>csides a small (juaiitity of vapor, from 1 to 3 parts in 
a thousand of carl)onic acid gas, and a trace merely of ammoniacal gas. 

b. Anthracite coal, charcoal, plumbago, coke, &c., have no other 
constituent than carbon ; they arc combined, to a small extent, with 
forein^ii matters, such as iron, silica, sulphur, alumina, &c. 

c. The constituents of woods, grains, &c., are given per cent., with- 
out regard to the foreign matters {metallic) which they contain. In 
oak^ cht'stmitf and Xor'wai/ pine, the ashes amount to about ^jj of 1 per 
cent., and in ash and mapln to y-^ of 1. In anthracite coals, at an 
average, they are about 7 per cent. 

(1. Fibrin, Gclotin, Albumen — Pro.vinialc animal constituents — 
Nutritious properties of animal matter. 

Fibrin is the basis of the muscle (loan meat) of all animals, and is 
also a large constituent of Uie blood. 

Gelatin exists largely in the skin, cartilages' ligaments, tendons and 
bones of animals. It also exists in the muscles and the membranes. 

Albumen exists in the skin, glands and vessels, and in the sennn of 
the blood. It constitutes nearly the whole of the white of an Q^g. 



CONSTITUENTS OP BODIES, 89 

The relative quantities by volume of the several gases going to 
constitute any particular compound, are readily ascertained by help 
of their respective specific gravities, compared with their relative 
weights, as given per cent, in the preceding table: — thus, the sp. 
gr. of hydrogen is .0689, and that of oxygen 1.1025, and 1. 102.3 -J- 
,0689 = 16 ; showing the weight of the latter to be 16 times that of 
the former per equal volumes, or, relatively, as 16 to 1. The per 
cent, by weif^ht, as shown by the table, in which these two gases 
combine to form water, for instance, is 11.1 and 88.9 ; or 11.1 of 
hydrogen and 83.9 of oxygen in 100 of the compound ; or as 88.9 -r- 
11. 1, — as 8tol: 16 — 8 = 2: two volumes, therefore, of the 
lighter gas (hydrogen) combine with one of oxygen to form water. 
Water, consequently, is a Protoxide of Hydrogen. 

Upon the principle of atomic weights — primal quantities, by 
weight, in which bodies combine, based upon some fixed radix, usually 
hydrogen as it forms with water, and as 1, — we have, for water, — 
H^ -j- ^^ = ■^^' ^- -^^ atom of -hydrogen, therefore, is 1, an atom 
of oxygen 8, and an atom of water 9. 

By the same rule as the preceding, the constituents of atmospheric 
air are found to be to each other in volume as 4 to 1, — 4 volumes of 
nitrogen and 1 volume of oxygen = atmospheric air. The weight 
of nitrogen to hydrogen per equal volumes, is as 14.14 to 1, Atomic- 
ally, therefore, oxygen being 8, it is as 7.07 to 1 ; hence we have 
N^ -|~ ^ = 36.28, the atomic weight of atmosphere. 

The vast condensation of the gases which takes place, in some in- 
stances, in forming compounds, may be conceived of, and the process 
for determining the same exhibited by a single illustration. We will 
take, for example, water. A single cubic inch of distilled water, at 
60°, weighs 252.48 grains. Its weight is to that of dry atmosphere, 
at the same temperature, as 827.8 to 1. A cubic inch of dry atmos- 
phere, therefore, at that density, weighs .305 of a grain. Hydrogen, 
we find by the table of Specific Gravities, weighs .0689 as much 
as atmosphere, and oxygen 1.1025 as much. A cubic inch of hydro- 
gen, therefore, weighs .0689 X .305 = .0210145 of a grain, and 
a cubic inch of oxygen 1.1025 X .305, = .3362625 of a grain. 
The constituents of water' bv volume are 2 of the first mentioned ffas 
to 1 of the latter; and .0210145 X 2 + .3362625 = .3782915 of a 
grain, = weight of three cubic inches of the uncondensed compound, 
J of which, .1260972 of a grain, is the weight of a volume 1 cubic 
inch. 

As the weight of a given volume of the uncondensed compound, is 
to the weight of an equal volume of the condensed compound, so are 
their respective volumes, inversely: then — 

.1260972 : 2512.48 :: 1 : 2002.28, the number of cubic inches of the 
two gases condensed into 1 inch to form water ; a condensation of 
2001 times. Of this volume of gases, |, or 1334.84 cubic inches, is 
hydrogen ; the remaining thirds 667.42 cubic inches, is oxygen. 

8*- 



90 PROPERTIES, ETC., OF BODIES. 

The foregoing method, though strictly correct, does not exhibit in a 
general way the most expeditious for solving questions of that nature, 
the condensation which takes place in the gases on being converted 
into solids, or dense compounds. It was resorted to, in part, as a 
means through which to exhibit principles and proportions pertaining 
thereto. 

As before ; one cubic inch of water weighs 252.48 grains, ^ of 
which, or 28.05-[- grains, is hydrogen, and ^^ or 224. 43 — grains, is 
oxygen. The volume of 1 grain of oxygen is 2.91-\- cubic inches, and 
the volume of hydrogen is 10 times as much, or 47.58-^ cubic inches. 
Therefore, 28.05 X 47.58 = 1334.G2, and L^M. 13 X 2.97 = 065.56, 
3= 2001.18, condensation, as before. 



Properties of the simple substances, and some of their compounds, not 
given in the foregoing. 

Bromine, — at common temperatures, a deep reddish-brown vola- 
tile liquid ; taste caustic ; o<lor rank ; boils at 110^ ; congcjiU at 4^ ; 
exists in sea-water, in many salt and mineral springs, and in most 
marine plants ; action upon the animal system very energetic and 
poisonous — a sinple drop placed ujx)n the beak of a bird destroys the 
bird almost instantly. A lighted taper, envelo|Mxl in its fumc^, burns 
with a llame green at the base and red at the top ; powdered tin or 
antimony brought in contact is instantly inflamed ; potash is exploded 
with violence. 

CiiLORiNK, — a greenish-yellow, dense gas ; taste astringent; odor 
pungent and (lisagreeal)le ; by a pressure of 60 lbs. to the s<iuare inch 
is reduced to a litjuid, and thence, by a reduction of the temperature 
below 32^, into a solid. It exists largely in sea-waler — is a constit- 
uent of common salt, and forms compounds with many minerals; is 
deleterious, irritating to the lungs, and corrosive ; has eminent 
bleiicliing properties, and is the greatest disinfecting agent known ; 
a lighted taper immersed in it burns with a red flame ; pulverized 
antimony is inflamed on coming in contact, so is linen saturated with 
oil of turpentine ; phosphorus is ignited by it, and burns, while im- 
mersed, with a pale-green flame; with hydrogen, mixed measure for 
measure, it is highly explosive and dangerous. 

Fluorine, — a gas, similar to chlorine, — exists abundantly in 
fluor-spar. 

Oxygen, — a transparent, colorless, tasteless, ino(li.(.uir>, iinn»vn»us 
gas ; supports respiration and combustion, but will not sustain life for 
any length of time, if breathed in a pure state. It is by far the most 
abundant substance in existence ; constitutes ^ of the atmosphere ; 



PROPERTIES, ETC., OF BODIES. 91 

I of water ; and nearly the whole crust of the earth is oxidized sub- 
stances. For further combinations and properties, see tables of Ele- 
mentary Constituents and Chemical Elements. 

Iodine, — at common temperatures, a soft, pliable, opaque, bluish- 
black solid ; taste acrid ; odor pungent and unpleasant ; fuses at 225^ ; 
boils at 347^ ; its vapor is of a beautiful violet color ; it inflame* 
phosphor^is, and is an energetic poison ; exists mainly in sea-weeds 

and sponges. 

Hydrogen, — a transparent, colorless, tasteless, inodorous, innox- 
ious gas ; if pure, will not support respiration ; if mixed with oxy- 
gen, produces a profound sleep ; exists largely in water ; is the basis 
of most liquids, and is by far the lightest substance known; burns in 
the atmosphere with a pale, bluish light ; mixed with common air, 1 
measure to 3, it is explosive ; mixed with oxygen, 2 measures to 1, 
it is violently so. 

Nitrogen, or Azete, — a transparent, colorless, tasteless, inodorous 
gas ; will not support respiration or combustion, if pure ; exists 
largely as a constituent of the atmosphere — in animals, and in fun- 
gous plants ; is evolved from some hot springs ; in connection with 
some bodies, appears combustible. 

Carbon, — the diamond is the only pure carbon in existence ; pure 
carbon cannot be formed by art ; charcoal is 97 per cent, carbon ; plum- 
bago, 95 ; anthracite, 93. Carbon is supposed by some to be X\\e hard- 
est substance in nature. A piece of charcoal will scratch glass ; but 
it is doubtful if this is not due to tlie form of its crystals, rather than 
to the first mentioned quality. It is doubtless the most durable. For 
combinations, &c., see table. 

Boron, — a tasteless, inodorous, dark olive-colored solid. 

Silicon, — a tasteless, inodorous solid, of a dark-brown color ; 
exists largely iu soils, quartz, flint, rock-crystal, &c. ; burns readily 
in air — vividly in oxygen gas ; explodes with soda, potassa, barryta. 

Phosphorus, — a transparent, nearly colorless solid, of a wax- 
like texture; fuse's at 108^, and at 550*^ is converted into a vapor; 
exists mainly in bones — most abundant in those of man — is poison- 
ous ; at common temperatures it is luminous in the dark, and by fric- 
tion is instantly ignited, burning with an intense, hot, white flame ; 
must be kept immersed in water. 

Selenium, — a tasteless, inodorous, opaque, brittle, lead-colored 



92 FROPEHTIES, ETC., OF TIODTES. 

Bolid, in the mass; in powder, a deep-red color; becomes fluid at 
216^, boils at C50^ ; vapor, a deep yellow ; exists but sparingly, 
mainly in combination with volcanic matter; is Ibuml in small (juan- 
tities combined with the ores of lead, silver, copper, mercury. 

Ammoniacnl gas, — N -f- IP; transparent, colorless, bi^^hly pun- 
gent and stimulating; ; alkaline ; is converted into a tninsparcnt liquid 
by a pressure of 0.5 atmospheres, at 50^ ; does not support respin- 

tion ; is inllammable. 

Carbonic acid gas, — C -f- (7^ ; transparent, colorless, inodorous, 
dense ; is converted into a liquid by a pressure of 30 atmospheres ; 
exists extensively in nature, in mines, deep wells, pits; is evolved 
from the earth, from ordinary combustion, especially from the combus- 
tion of charcoal, and from many mineral sprinfrs ; is expired !)y man 
and animals; f^miis 44 jx^r cent, of the carbonate of lime called mar- 
ble ; the brisk, sparklinj; appearance of soda-water, and most mineral 
waters, is due to it.s pri*scnce. It is neither a combustible nor a sup- 
porter of combu.-r^lion ; and, when mixed with the atmosphere to an 
extent in which a candle will not burn, is di^tructive of life. IJcinp 
heavier than atmosphere, it may l)c drawn up from wells in large o|>ef> 
buckets ; or it may be expelled by cxplodin,9r ^nmpowdcr near the b<»t 
tom. Large quantities of water thrown in will absorb it. 

The above pas is expired by man to the extent of 1630 cubic inches 
per hour ; it is i^cnorated by the burninj^ of a wax candle to the ex- 
tent of BOO cubic inches per hour: and, by the burnin;: of**Ow>- 
phcnr^'" (in the production of lijtjht equal to that aHonled by 1 wax 
caM(ll(%) to the extent of 875 cubic inches per hour. Two burning 
candles, thereffv-r, -i^i'''* ii>" -'ir in -iluuit \\^q o^.i^^o ovt...,t ..« t |w.r_ 
son. 

Carbonic oxide gas, — C -f- O ; transparent, colorh^ss, insipid; 
odor offensive ; does not support combustion ; an animal confinetl in 
it soon dies; is hi^rhly inllammable, burninn^ with a pale blue flame; 
mixed with oxygen, 1 to 2, is explosive — with atmosphere, even in 
small quantity, is productive of j^iddiness and fainting. 

Carbureted hydrogen gas, — C -|- H'^ ; transparent, colorless, taste- 
less, nearly inodorous ; exists in marshes and stagnant pools — is 
there formed by the decomposition of vegetable matter ; extingui.sbes 
all burning bodies, but at the same time is itself highly combuslibie, 
burning with a bright but yellowish flame ; it is destructive to life, if 
respired. 

Cyanogen — Bicarburet of Nitrogen — a gas, — N -|- L , i;.iii&- 
parent, colorless, highly pungent and irritating , under a pressure of 



PROPERTIES, ETC., OF BODIES, 93 

3.6 atmospheres, becomes a limpid liquid ; burns with a beaatiful 

purple flame. 

Hydrochloric add gas — Muriatic acid gas, — H -[" d* (chlorine) ; 
transparent, colorless, pungent, acrid, suffocating ; strong acid taste. 

Nitrous oxide gas — Protoxide of Nitrogen, ^* laughing gas,^^ — 
N -f- O ; transparent, colorless, inodorous ; taste sweetish ; powerful 
stimulant, when breathed, exciting both to mental and muscular ac- 
tion ; can support respiration but from 3 to 4 minutes ; is often per- 
nicious in its effects. 

Nitric oocide gas — Binoxidc of Nitrogen, — N -j- O^ ; transparent, 
colorless ; wholly irrespirable ; lighted cliarcoal and phosphorus burn 
in it with increased brilliancy. 

Olefiant gas — Bicarlmreted hydrogen gas — ^* coal gas,^^ — C* -|- 
H^ ; transparent, colorless, tasteless, nearly inodorous, when pure ; 
does not support respiration or combustion ; a lighted taper inunersed 
in it is imm.ed lately extinguished. It burns with a strong, clear, 
white light ; mixed with oxygen, in the proportion of 1 volume to 3, 
it is highly explosive and dangerous. 

Phosphureted hydrogen gas, — P -|- H^ ; colorless ; odor highly 
offensive ; taste bitter ; exists in the vicinity of swamps, marshes, 
and grave -yards ; is formed by the decomposition of bones, mainly ; 
is highly inflammable ; takes fire spontaneously on coming in contact 
with the atmosphere ; mixed with pure oxygen, it explodes. It is 
the veritable *' Will o' the wisp," 

Sulphureted hydrogen gas — Hydrosulphnric add gas, — S -|- H ; 
transparent, colorless ; taste exceedingly nauseous ; odor offensive 
and disgusting ; is furnished by the salphurets of the metals in gen- 
eral — also by filthy sewers and putrescent Qggs. It is very destruc- 
dve to life ; placed on the skin of animals, it proves fatal. It burns 
with a pale blue flame, and, mixed with pure oxygen, it is explosive. 

Hydrocyanic add — Prussic add, — N -}- C^ 4" -^ » ^ colorless, 
limpid, highly volatile liquid ; odor strong, but agreeable — similar 
to that of peach-blossops ; it boils at 79^^ and congeals at ; exists 
in laurel, the bitter ^almond, peach and peach kernel. It is a most 
virulent poison, — a drop placed upon a man's arm caused death in a 
few minutes. A cat, or dog, punctured in the tongue with a needle 
fresh dipped in it, is almost instantly deprived of life. 

Hydrofluoric add, — F -}- H ; a colorless liquid, in well stopped 
lead or silver bottles, at any temperature between 32° and 59®. It is 



91 PROP/IKTICS, ETC., OF DODIIS. 

obtained by the action of sulfiliuric acid on fluor-s>par. It readily 
acts upon and is used for eichint^ on glass. It is the most destructive 
to aniitial matter of any known substance. 

NitrohydrochJoric add — ** aqua regia,^^ — (1 part nitric acid and 4 
parts muriatic acid, by measure ;) — a solvent for gold. The best sol- 
vent for gold is a solution of sal ammoniac in nitric acid. 

Nitrosulphuric acid, — (1 part nitric acid and 10 parts sulphuric 
acid, by measure) — a solvent for silver ; scarcely acts upon gold, 
iron, copper, or lead, unless diluted with water ; is used for separat- 
ing the silver from old plated ware, &c. The Ix^st solvent for silver, 
and one which will not act in the least upon gold, copper, iron, or 
lead, is a solution of 1 part of nitre in 10 parts of concent rat('^ 
phuric acid, by weight, heated to 100'. This mixture will di 
about J its wei;,Hit of silver. The silver may be recovered by adding 
common salt to the solution, and the chloride decomposed by the car- 
bonate of soda. 

Sclenic arid, — Se -j^ ()^ ; obtained by fusing nitrate of potassa with 
selenium — a solvent tor gold, iron, copi>er, and zinc. 

Silicic acid, — (Silicn — silex ; base Silicon) — Si 4" O* ; exists 
largely in sand. Common glass is fused sand and protoxide of potas- 
sium (carl)onate of potassa — notash) in tlie proportion of 1 part by 
weight of the former to 3 of the latter. 

Manganese, compounded with oxygen, in different proportions, im- 
parts the various colors and tints given to fancy glass ware, now tm 
generally in vogue. 



} 
I 



SECTION III. 
PRACTICAL ARITHMETIC. 



VULGAR FRACTIONS. 

A fraction is one or more parts of a Unit. 

A vulgar fraction consists of two terms, one written above the 
other, with a line drawn between them. 

The term below the line is called the denominator, as showing the 
denomination of the fraction, or number of parts into which the unit 
is broken. 

The term above the line is called the numerator, as numbering the 
parts employed. These together constitute the fraction and its 
value. 

A vulgar fraction always denotes division, of which the denomina- 
tor is the divisor and the numerator the dividend. Its value as a unit 
is the quotient arising therefrom. 

A simple fraction is either a proper or improper fraction. 

A proper fraction is one whose numerator is less than its denomina- 
tor, as ^, f , f i-, &c. 

An improper fraction has its numerator equal to or greater than its 
denominator, as f , ^, f |-, &c. 

A mixed fraction is a compound of a whole number and a fraction, 
as 1^, 5i^, 12^3^, &c. 

A compound fraction is a fraction of a fraction, as J of | ; | of 
I of J-f , &c. 

A complex fraction has a fraction for its numerator or denom- 



i 4 i- 5i 

3 ' 3' 
5 T 



inator, or both, as -> t» |» ""? ^^-t ^^^ i^ ^^^^ 2 "^ ^ » "^ "^ f » 
3 TT a: 4 



J-i-l; 51-1.4, &e. 

REDUCTION OF VULGAR FRACTIONS. 
To reduce a fraction to its lowest terms. 

This consists in concentrating the expression without changing the 
value of the fraction or the relation of its parts. 

It supposes division, and, consequently, by a measure or measures 
common to both terms. 

It is said to be accomplished when no number greater than 1 will 
divide both terms without a remaindei : — therefore, 



96 VULGAR TKACnONS. 

Rule. — Divide both terms by any number that will diride then 
without a remainder, and the quotient again as before ; continue so to 
do until no number greater than 1 will divide them, — or divioe by 
the greatest comnwn measure at once. 

Example. — Reduce j^^ to its lowest terms. 
4)T¥T^ = lii^2 = |S|--9 = H^3 = f. An,. 

To reduce an improper fraci ion to a mixed or xckole number, 

RuLC. — Divide the numerator by the denominator and lo the 
whole number in the quotient annex the remainder, if any, in iom\ of 
a fraction, making the divisor the denominator as before ; then reduce 
the fraction to its lowest terms. 

Example. i=li; -}|=1A = 1J; t1 = 2. 
To reduce a mixed fraction to an equivaknt rmpnypcr fraction. 

Rule. — Multiply the whole number by the denominator of the 
fractional part, and to the product add the numerator, aod place ikeir 
sum over the said denominator. 

Example. ^- Reduce 3^ and 12| to improper fractions. 
3X4 = 12-f-l = J^. Ans. 12 X -f 8 = -4^. Ans. 

To reduce a whole number to an equivalent fraction having a givem 

deninninator. 

Rule. — Multiply the whole numbet by the given denominator, 

and place the said denominatt^r under the product. 

P^XAMPLE. — How may 8 be converted into a fraction whose de- 
nominator is 12 ? 

8X12=»?S. Ans. 

To reduce a compound fraction to a simple one. 
. Rule. — Multiply all the numerators togcihcr for a numerator, and 
all the denominators lofrethcr for a denominator ; the fniction thus 
formed will be an equivalent, Init oftcii not in its lowrst terms. Or, 
concentrate the expression, when practicable, by recipn)cally expung- 
ing, or writing out, such factors as exist or are attainable common to 
both terms, and then multiply the remaining terms as directed above. 

NoTB. — This last practice is called cancellation, or cantelling the terms. It conststt^ 
as has been staled, in reciprocally annulling, or cas^iin? om. equal Talttea from l>oih terms^ 
whereby the expression is conf eiUrale*!, and ihe relation of the parla kept unJistfirbed ', 
and il may always be carried lo the extent of reducin? the fraciioii to its lowest ternM, 
before any mnltiplicalion, as Goal, is resorted lo ; and often, therefore, to the ext«nt that 
vuch multiplication is inadmissible, the terms havii^g beea caiic«il«l by each other uuli) 
bet B single number is left ia each. 



Operation by cancellation, ^ ^ ^ = ^. Ans, 



VULGAR FRACTIONS. 97 

ExAMPLK. — Reduce § of | of ij to a simple fraction. 
Operation by multiplication, §X fXi-==^^5- = |-- -4.7tf. 

Example. — Reduce f of f of ^^ of | of |- of 2 to a simple 

Fraction. 

By multiplication, f X f X -V- X f X f X | = fffl = i- ^^• 
The last example stated ) 2 3 12 6 5 2 
for cancellation, J 3 4 8 8 9 

PROCESS OF CANCELLING THE ABOVE. 

1. The 3 in num. eqyalg the 3 in denom., therefore erase both. 

2. The first 2 in num. equals or measures the 4 in denom. twice, therefore place a 2 
under the 4, and erase the 4 and 2 which measured it — (as 4 : 2 : : 2 : 1.) 

3. The 2 (remaining factor of 4 and 2 erased) in denom., and the remaining 2 in num., 
will cancel each other, — erase Ihem. 

4. The 12 and 6 in num. = 72, and the 9 and 8 in denom. = 72 ; these, therefore, ia 
their relations as factors equal each other, and may be erased. 

The remaining factors represent the true value of the compound fraction, and will ba 
found = I, as by multiplication. 

Example. — Reduce \% of -^^ to a simple fraction. 

3 

3 

^^ X ^,. Or ^-^^^(=18 4-6, andl2-J-6)=§XA 

13 xa:^ '"''13x3:^ =1^. aus. 



To reduce fractions of different denominations to an equivnlent simple 
one, — to a fraction having a common denominator. 
Rule. — Multiply each numerator by all the denominators except 
its own and add the products together for the numerator, and multiply 
all the denominators together for a denominator. 

Note. — Whole numbers «nd fractions other than simple, must first be reduced to sim- 
ple fractious before they can be reduced to a fraction having a common denominator. 

Example. — Reduce | and | to an equivalent simple fraction. 
Example. — Reduce ^, -f, |-, and ^ to an equivalent. 
9 



98 TULGAE FRACTlOPrS. 

To reduce a complex fraction to a simple one. 
Rule — Multiply the numerator of the upper fraction by the 
denominator of the lower, for the new numerator ; and the denomi- 
nator of xhe upper by the numerator of the lower for the new denom- 
inator. 

141 5; 

Examples. — Reduce -, -, g, and — '- each to a simple fractioo. 

V, and V.X k = Y. = H' ^^' 

To reduce Vulgar Fractions to Cfpiivalent Decimals. 

Rule. — Divide the numerator by the denominator ; the quotient is 
the decimal, or the whole number and decimal, as the case may be. . 

Example. — Reduce J, 4^, -f^-, to decimals. * 

7 -^ 8 = 0.875; 4^ = ^^ = 4'.6 ; 14-^12=1.166+. Ans. 

To find the greatest common measure or divisor of both terms of a simple 
fraction y or of two nitmbcrs. 

Rule. — Divide the greater numl)er by the less ; then divide the * 
divisor by the remainder ; and so on, oontinninjj to divide the last 
divisor by the last remainder until nothint^ remains ; the last divisor 
is the greatest common measure of the two terms. 

Example. — What is the greatest common measure of ^^ J or of 
132 and 250 ? .* 

132 ) 256 ( 1 I 

132 I 

124 ) 132 ( 1 I 

124 I 

8 ) 124 ( 15 I 

120 « 

4)8(2 ' 

8 



4. Ans. 



To find the least common denominator of two or more fractions of dif- 
ferent denominators y Or the least common multiple of two or more 
numbers. 

Rule. — Divide the given denominators, or numbers, by any num- 
ber greater than 1, that will divide at least two of them without a 
remainder, which quotient together with the undividecl numbers set 
in a line beneath. Divide the second line as tx^fore, and so on, uotil 



VULGAR FRACTIONS. 99 

there are no two numbers in the line that can be thus divided ; the 
product of all the divisors and remaining numbers in the last (undi- 
vided) dividend is the least common denominator, or multiple sought. 

Example. — What is the least common denominator of ^ij? 2^> 
and 52^, or of 20, 25, and 50 ? 

5 ) 20.25.50 
2) 4.5.10 



5 ) 2.5. 5 
2. 1 .T 
5X2X5X2= 100. Ans, 

ADDITION OF VULGAR FRACTIONS. 

Sum of the products of each numerator with all the denominators except that of iho 
numerator involved, forms numerator of sum. 
Product of all the denominators forms denominator of sum. 

Rule. — Arrange the several fractions to be added, one after 
another, in a line from left to right ; then multiply the numerator of 
the first by the denominator of the second, and the denominator of the 
first by the numerator of the second, and add the two products 
together for the numerator of the sum ; then multiply the two denom- 
inators together for its denominator ; bringdown the next fraction, and 
proceed in like manner as before, continuing so to do until all the 
fractions have been brought down and added. Or, reduce all to a 
common denominator, then add the numerators together for the 
numerator of the sum, and write the common denominator beneath. 

Examples. — Add together |, |, |, and ^. 

i = f + l = l = if.and| + f = f=f|,andit+fl = fl 

SUBTRACTION OF VULGAR FRACTIONS. 

Product of numerator of minuend and denominator of subtrahend, forms numerator of 
minuend, for common denominator. 

Product of numerator of subtrahend and denominator of minuend, forms numerator of 
subtrahend, for common denominator. 

Product of denominators forms common denominator. 

Ditference of new found numerators forms the numerator, and common denominator the 
denominator, of the difference, or remainder sought. 

Rule. — Write the subtrahend to the right of the minuend, with 
the sign ( — ) between them ; then multiply the numerator of the 
minuend by the denominator of the subtrahend, and the denominator of 
the minuend by the numerator of the subtrahend ; subtract the latter 
product from the former, and to the remainder or difference affix the 



100 VtlLGAR FRACTIONS. 

product of the two denominators for a denominator; the sum thui 
formed is the answer, or true difference. 

Examples. — Subtract ^ from |, also ^ from \^, 

1|«3_ 55-51=^4^. Arts. 

DIVISION OF VULGAR FRACTIONS. 

Product of numerators of dividend and denominators of divisor, forms numerator of 
quotient. 

Product of denominators of dividend and numerators of divisor, forms denominator of 
quotient; therefore, 

Rule. — Write the divisor to the ri^ht of the dividend with the 
sig-n (-r-) between them ; then multiply the numerator of the dividend 
by the denominator of the divisor, for tlie numerator of the quotient, 
and the denominator of the dividend by the numerator of the divisor, 
for the denominator of the quotient. Or, invert the divisor, and mul- 
ti])ly as in multiplication of fractions. Or, proceed by cancellation, 
when practicable. 

ExAMPLKs. — Divide ^ by J ; | by ^ ; ^ by [^ ; and J of | of 
I of I by 1 of lof |of §. 

• i^.? = |; f-^i = |; *H"H = J5; ori;^ii = ii Arts. 

iX3X^Xi = iV^ = fV, and ixix|xl=^\ = TV. 

and-rV ~TV = f^ = -^3^-6|. Ans. 

FORM FOR CANCELLATION. — EXAMPLE LAST GIVEN. 

1354 4243 20 

2 4 6 3 1 1 3 2 = 1' '4.5., as above. 

Note. — The fi^resioini: eximplc can bo cancelled to the extent of leaving but a 4 and a 
5 (= 2()) nuinerattirs. aiul a :$ deaominaior. Units, or I's, in the expressions, are ralua. 
les:j, as a sum mulliplied by 1 is not incrc;isod. 

MULTIPLICATION OF VULGAR FRACTIONS. 

Product of nunieralors of niultiplier and multiplicantl, forms numerator of product. 
Product of denominators of multiplier and multiplicand, forms denominator of product. 

Rule. — Multiply the numerators together for a numerator, and tlie 
denominators together for the denominator. 

Examples. — Multiply | by J ; | by 7 ; |i by -^ ; ^ of § of | 
by I of i off. 

iXi = J; 3Xl = ^:rM |4X-V^=W = -V-; iX §X| 
= /j = i , and I X 1 X i = :>^ = i and 4 X J = tV ^'" 



VTTI/JAR FRACTIONS. 101 

MUI.TIPLICATION AND DIVISION OF FRACTIONSj COMBINED. 

It has been seen that a compound fraction is converted into an 
equivalent simple one, by multiplying the numerators together for a 
numerator, and the denominators together for a denominator ; and 
it has also been seen that a series of simple fractions are con- 
verted into a product, by the same process. It is therefore evident 
that compound fractions and simple, or a series of compound and a 
series of simple, may be multiplied into each other, for a product, by 
multiplying all the numerators of both together for a numerator, and 
all the denominators of both together for a denominator ; and that the 
product will be the same as w^ould be obtained, if the compound were 
first converted into an equivalent simple fraction, and the simple frac- 
tions into a product or factor, and these multiplied together for a 
product. 

It has also been seen that a fraction is divided by a fraction by mul- 
tiplying the nun^terator of the dividend by the denominator of the 
divisor, for the numerator of the quotient, and the denominator of the 
dividend by the numerator of the divisor, for the denominator of the 
quotient ; and that this multiplication becomes direct as in multiply- 
ing for a product, if the divisor is inverted. And it is clear that a 
compound divisor, or a series of simple divisors, or both, may be used 
instead of their simple equivalent, and with the same result, if all are 
inverted. 

It is therefore evident that any proposition, or problem, in fractions, 
consisting of multiplications and divisions both, and these only, no 
matter how extensive and numerous, or whether in compound frac- 
tions, or simple, or both, may be solved, and the true result obtained, 
as a product, by simply multiplying all the numerators in the state- 
ment together for a numerator, and all the denominators in the state- 
ment for a denominator, all the divisors in the statement being 
inverted ; that is, all the numerators of the divisors being made denom- 
inators in the statement, and all the denominators of the divisor being 
made numerators in the statement. And it is further evident that a 
proposition stated in jthis way, admits of easy cancellation as far as 
cancellation is practicable, which is often to great extent. 

Example. — It is required to divide 12 by § of 1 ; to multiply the 
quotient by the product of 4 and 8 ; to divide that product by |- of ^ 
of 8 ; to multiply the quotient by J of f of ^ ; and to divide that 
product by the product of 5 and 9. 

9* 



102 VULGAR FRACTIONS. 

STATEMENT. 
(Diyidends read from right to left, divisors from left to right) 



Numerators of dividends and denominators of divisors. 



B a 



Numerator of ^ ejco^oo^o^o t-aoo S Dividend of 

Btalcmcrit. S •-< f Btalcineiit. 



Denominator ) C) CO l^ co CD 00 co ^ o Ci c Divisor of 
of statement. S ._^^ , ^^ !^^ J sialemenL 

'sj08{Aipjo fuoicJ9ianu pmi epuopiAipjo sjoicu;uioti3(j 



The ansiccr to the above proposilion is 1J-|, and the proposition 
as stated may be readily cancelled to its lowest terms. It may be 
cancelled to the extent of leavin^x but 4, 4, 2 in tlie immerator, and 7, 
3, in the denominator, ^^^- = i'i = IH- 

To reduce a fraction in a hii^hcr denomination to an equivalent fraction 
in a given lower denomination. 
KuLE. — IMultiply the fraction to be reduced — numerators into 
numerator and denominators into denominator — by a fraction whose 
numerator represents the number of parts of the lower denomination, 
required to make one of the denomination to be reduced. 

Example. — Reduce J of a foot to an equivalent fraction in inches. 

Example. — Reduce ^ of a jiound to an equivalent fraction in | 
ounces. 

I X Y- = -V- -^ 3 = ^V- = ¥• ^rt,. 
Or, ^ X -V- X a = ^ii = 2^a. Ans. 

To reduce a fraction in a lower denomination to an equivalent fraction 
in a givrn higher daiomination. 

Rule. — Multiply the fraction to be reduced — numerator into 
denominator and denominator into numerator — by a fraction whose 
numerator represents the number of parts required of the lower 
denomination to make 1 of the higher. 

Example. — Reduce ^>^- inches to an equivalent fraction in feet. 
^^^ = U = \- '-^ris. Or, V X tV = f i = *• ^^- 



ii 



VTJLGAE FRACTIONS. 103 

Example. — Reduce ^- two third ounces to an equivalent frac- 
tion in pounds. 

^X| = -V<'-H-Jt& = |»=-|. Ans. 

0r,4j<iX|XTV = ll = i- ^««- 

To reduce a fraction in a higher to whole numbers in lower denomi- 
nations. 
Rule. — Multiply the numerator of the given fraction by the num- 
ber of parts of the next lower denomination that make- one of the 
given fraction, and divide the product by the denominator. Multiply 
the numerator of the fractional part of the quotient thus obtained by 
the number of parts in the next lower denomination that make 1 of 
the denomination of the quotient, and divide by its denominator for 
whole numbers as before ; so proceed until the whole numbers in each 
denomination desired are obtained. 

Example. — How many hours, minutes, and seconds, in -^^ of a 
day? 

^9^X24 =_2^1_6 = 15^ 3 X 60 = X|a= 25, -^X 60 = 50(1 = 42 6^ =« 

15 h., 25 m., 42|- sec. Ans. 
Example. — How many minutes in -^^ of a day ? 

_9^ X 2 4 X 6 =_L2_9_6P_ = 925^. AnS. 

To reduce fractions , or whole numbers and fractions^ in lower denomi- 
nations^ to their value in a higher denomination. 
Rule. — Reduce the mixed numbers to improper fractions, find 
their common denominator, and change each whole number and 
numerator to correspond therewith. Then reduce the higher numbers 
to their values in the lowest denomination, add the value in the lowest 
denomination thereto, and take their sum for a numerator. Multiply 
the common denominator by the number required of the lowest denom- 
ination to make one of the next higher, that product by the number 
required of that denomination to make 1 of the next higher, and so 
on, until the highest denomination desired is reached, and take the 
product for a denominator, and reduce to lowest terms. 

Example. — Reduce 5| oz., 3^ dwts., 2^ grs., troy, to lbs. 
Y . 1^ . 5 =:i&o^9^6^L^ . therefore, 

160 X 20 = 3200 

\ 96 



3296 X 24 = 79104 
75 



79179 



30 X 24 X 20 X 12 = 172800 



I =.458 4- lbs. Ans. 



104 DECIMAL FRACTIONS. 

Example. — Reduce 11 hours, 59 minutes, 60 seconds, to the frao- 
tion of a day. 

11X60*= 660 
59 

719 X 60 = 43140 
60 



43200 ) , , 

> =» i. Arts. 

60 X 60 X 24 = 86400 ) 

Example. — Reduce 15 h., 25 m., 42f^ sec, to the fraction of a 
day. 

15 X 60 X 60 = 54000 
25X60= 1500 
42f 

55542f 

7 



7 X 60 X 60 X 24 = 004H00 



388800 , 

Ans. 



!=,v 



To work fractions, or whole numlnrs and fractions, by the Rule of 

Three, or Proportion. 

Rule. — Reduce the mixed terms to simple fractions, state the 
question as in whole numbers, invert the divisor, and multiply and 
divide as in whole numbers. 

Example. — If 2i yards of cassimere cost $41, what will | of a 
'^ard cost? 2j = | ; 4j = J/- ; then, 

^: V-:: r. «^, = V-gfx? = W = Sl.27,5. Ans, 



DECIMAL FRACTIONS. 

A decimal fraction is written with its numerator only. Its denomi- 
nator is understood. It occupies one or more places of figures, and 
has a point or dot (.) prefixed or placed before it. The dot (.) alone 
distinguishes it from an integer or whole number. It supposes a 
denominator whose value is a unit broken into parts, having a ten- 
fold relation to the number of places the numerator occupies. The 
denominator, therefore, of any decimal is always a unit (1) with as 
many ciphers annexed as the numerator has places of figures. Thus, 
the denominator of .1, .2, ,3, &c., is 10, and the fractions arc read, 
one tenth, tivo tenths, three tenths, &c. The denominator of ,01, .11, 
.12, &c., is 100, and these are read, one hundredth, eleven hundredths, 



DECIlVtAL FRACTIONS. 105 

twelve hundredths, &c. The denominator of .001, .lOl, .125, &c., is 
1000, and these are read one thoiisandth , one hundred and one thousandths, 
one hundred and twenty-five thousandths, &c. The denon^inator of a 
decimal occupying four places of figures as .7525 is 10000, and so on 
continually. 

The first figure on the right of the decimal point is in the place of 
tenths, the second in the place of tenths oi tenths, or hundredths, the 
third in the place of tenths of tenths of tenths, or thousandths, &c. 
Thus the value of a decimal occupying four places of fio^ures, as 

, . 7525 752i 75i U 4- 

•7525, for example, is , = , = , = — ^ -| ^- = 

__^____ ^ 10000 1000 100 10 ' 100 

i i 

1 — . A decimal is converted into a vulg-ar fraction of 

1 ' 100 ^ 

equal value, by affixing its denominator. 

Ciphers placed on the right of decimals do not change their value. 
Thus, .1850 = .185, plainly for the reason that the denominator of 
the latter bears the same relation to that of the former that 185 bears 
to 1850 ; from both terms of the fraction a ten fold has been dropped. 

Ciphers placed on the left of decimals decrease their value ten fold 
for every cipher so placed. Thus, .1 = yV? •^l ^^ tqts^ •^^^ ^^ 

A mixed number is a whole number and a decimal. Thus, 4.25 is 
a mixed number. Its value is 4 units, or ones, and -^-^ of 1, = 
^^^ = 4^. The number on the left of the separatrix is always a 
whole number — that on its right, always a decimal. 

ADDITION OF DECIMALS. 

Rule. — Set the numbers directly under each other according to 
their values, whole numbers under whole numbers, and decimals un- 
der decimals ; add as in whole numbers, and point off as many places 
for decimals in the sum as there are figures in that decimal occupying 
the greatest number of places. 

Examples. -- Add together .125, .34, .1, .8672. Also, 125, 34.11, 
.235. 1.4322. 



.125 

.34 

.1 

.8672 

1.4322 Ans. 



125. 
34.11 

.235 
1.4322 



160.7772 Ans, 



SUBTRACTION OF DECIMALS. 

Rule. — Set the numbers, the less under the greater, and in other 
respects as directed for addition ; subtract as in whole numbers, and 



106 



DECIMAL FRACTIONS. 



point off as many places for decimals in the remainder as the decimal 
having the greatest number of figures occupies places. 



Examples. — Subtract .2053 from .8. 

.8 
.2653 



.5347 Ans, 



Also, 11.5 from 238.131. 

238.134 
11.5 

220.634 Ans, 



MULTIPLICATION OF DECIMALS. 

Rule. — Multiply as in whole numbers, and point off as many 
places for decimals in the prod not as there arc decimal places in the 
multiplicand and multiplier h»)th. If the product has not so many 
places, prefix ciphers to supply the deficiency. 

Examples. — Multiply 14.125 by 3.4. Also, 5.14 by .007. 



14.125 
3.4 



50500 
42375 

48.0250 = 48.025. Ans. 



5.14 
.007 



.035U8 Ans, 



Note. — Multiplying by a decimal is equivalent to divuiin? by a whole number that 
bears ihc same rehilion lo :i unit ihal a unil )>e.irfl to a ilccini.il. Multiplying by a deci* 
I1>al, therefore, is p<juivalent m ilivjtljiiir by the iltMiorninator of a fririion of criual value 
whose numerator la 1. or of illvidin? by the <l«'n<»minaior of a fmciion of e^^ual value whofto 
numerator is more than I, and multiplying the (pjoticul by the numcmlor. Thus, Ihd 
decimal .2r^ = ^J>^ = {, and the decimal .875 = -rVuV = ^- ^"^ ^^^ X -25 =» 
3.5r>7.1. and 14.2;{ -J- 1 = 3 5:»75. So, ali^, 14.23 X •t>75 «= 12.45l2i", and 14.23 -i- 8 s=a 
1.77S75 X ^ = 12 43125. It is sometimes a saving of labor and mailer of convenience lo 
achieve multiplication by this process. 

DIVISION OF DECLMALS. 

Rule. — Write the numbers as for division of whole numbers, then 
remove the separatrix in the dividend as many j)laoes of fifrures to the 
rip^ht, (supplyinp- the places witli ciphers if they are not occupied,) aa 
there are decimal fifrures in the divisor; consider the divisor a whole 
number and divide as in division of whole numbers. 



Example. — Divide .5 by .17. Also, .129 by 4. 



.17).50(2.944-. A71S. 
34 

160 
153 

70 
68 



4).129(.032-f. 
12 

9 
8_ 

1 



Ans, 



DECIMAL FRACTIONS. 



107 



Examples. — Divide 16.5 by 1.232. Also, 1.2145 by 12.231. 



1.232,) 16.500, (13.39284-. 
1232 

4180 
3696 



Ans., 



4840 
3696 
11440 

11088 

3520 
2464 



12.231,)1.214,50(.09929+> . 

1 1 f\f\ "TO Anno )" -A-US 



1 100 79 
113 710 
110 079 



0993 



n- 



3 6310 
2 4462 

1 18480 
1 10079 

8401 



10560 
9856 

704 

Note. — Dividing by a decimal is equivalent to multiplying by a whole number that 
bears the same proportion to a unit that a unit bears to the decimal. Dividing by a deci- 
mal, therefore, is efjuivalent to multiplying by the denominator of a fraction of equal 
value whose numerator is 1, or multiplying by the denominator of a fraction of equal 
value whose numerator is more than 1, and dividing the product by the numerator. Di- 
viding by a fraction is equivalent to multiplying by its denominator and dividing the 
product by its numerator, or dividing by its numerator and multiplying the quotient by 
its denominator. Thus, .5= ^^^ = \, and .7o = ^^jj = }. And 12.24 -i- .5 = 24. 4S, 
and 12.24 X 2 = 24.4S. So, also. 12.24 -f- .75 =5 IG. 32, and 12.24 X -^ = ^^-^ -r 3 = 
16.32. This method of accomplishing division may often be resorted to with convenience. 



REDUCTION OF DECIINTALS. 

To reduce a decimal in a higher to ichole numbers in successive lower 
denominations. 

Rule. — Multiply the decimal by that number in the next lower 
denomination that equals one of the denomination of the decimal, and 
point off as many places for a remainder as the decimal so multiplied 
has places. Multiply the remainder by the number in the next lower 
denomination that equals 1 of the denomination of the remainder, and 
point off as before ; so continue, until the reduction is carried to the 
lowest denomination required. 

Example. — What is the value of .62525 of a dollar? 
.62525 



100 



Cents 



Mills, 



62.52500 
10 



5.25000 An, . 62 cents 5^ mills. 



108 DECIMAL FRACTIONS. 

Example. — What is the value of .46325 of a barren 





.46325 
32 


Gallons, 


14.82400 
4 


Quarts, 


3.296 
2 


Pints, 


.592 
4 



Gills, 2.368. Arts. 14 gals. 3 qt«. 2,^ ^ilk. 

Example. — IIow many pence in .875 of a pound? 
.875 X 210 --2 10. Ans. 

To reduce decimals, or whole numbers and decimals, in lower denomin* 
' afionSj to their value in a higher denomination. 
RuLK. — licMhire all the pivcn denominations to their value in the 
lowest denomination, then divide their sum hy the number required of 
the lowest denomination to make one of the denomination to which 
tlie whole is to be reduced. 

Example. — Reduce 14 gallons, 3 quarts, 2.368 pills, to the deci- 
mal of a barrel. 

14 X 4 = 56 + 3 = 59 X 8 = 472 + 2.368 = 474.368. 
8 X 4 X 32 = 1024 ) 474.368 ( .46325. Ans, 

To rrork (hcimah, or whole numbers and decimals, by the Rule of 
Three, or Proportion. 
Rule. — State the question and work it as in whole numbers, 
taking- care to point off as many places for decimals in the product to 
be used as tlie dividend, as there are decimals in the two terms which 
form it, and to remove the decimal point therein as many places to tho 
right as there are decimals in the term to 1k3 used as a divisor, before 
the division is had. 

Example. — If .75 of a pound of copper is worth .31 of a dollar 
how much is 3.75 lbs. worth? 

.75 : .31 :: 3.75 
^1 

375 
1125 



.75) 1.16,25 ($1.55. Ans. 



PROPORTION. 109 

PROPORTION, OR RULE OF THREE. 

The Rule of Proportion involves the employment of three terms 
— a divisor and two factors for forming a dividend — and seeks a 
quotient, which, when the proposition is written in ratio, bears the 
same relation to the third term that the second term bears to the first. 
Two of the terms given are of like name or nature, and the other is 
of the name or nature of the quotient or answer S'Jught. That of 
the nature of the answer is always one of the factors for forming the 
dividend, and, if the answer is to be greater than that term, the larger 
of the remaining two is the other ; but if the answer is to be less 
than that term, the less of the remaining two is the other — the 
remaining term is the divisor. 

Example. — If $12 buy 4 yards of cloth, how many yards will 
$108 buy? 

^ X 108 _ 108 

pZ — 36 yards. Ans, 

3 

Example. — If 4 yards of cloth cost $12, how many dollars will 
36 yards cost 1 

12 V ?6 

—9 — = 108 dollars. Ans. 
4 

Example. — If 30 men can finish a piece of work in 12 days, how 
many men will be required to finish it in 8 days? 

= 45 men. Aris. 

o 

Example. — If 45 men require 8 days to finish a piece of work, 
how many men will finish the same work in 12 days? 
45 X 8 



12 



30 men. Ans. 



Example. — If 8 days are required by 45 men to finish a piece 

of work, how many days will be required by 30 men to finish the 

same work? 

8X45 

— — — = 12 days. Ans. 

Example. — If 12 days are required by 30 men to perform a piece 

of work, how many days will be required by 45 men to do the same 

work? 

12X30 ^ , 

— — = 8 days. Ans. 

45 

Example. — I borrowed of my friend $150, which I kept 3 months, 
and, on returning it, lent him $200 ; how long may he keep the sura 

10 



110 



COMPOUND PROPORTION. 



that the interest, at the same rate per cent., may amount to that which 
his own would have drawn? 

150 X 3 -^ 200 = 2^ months. Ans. 

Example. — A garrison of 250 men is provided with provisions lor 
30 days, how many men must be sent out thai the provisions may last 
those remaining 42 days ? 

250 X 30 -7- 42 = 179, and 250 — 179 = 71. Arts, 

Example. — If to the short arm of a lever 2 inches from the ful- 
crum there be suspended a weight of 100 lbs., what power on the 
long arm of the lever 20 inches from the fulcrum will be required to 
raise it? 

20 : 2 :: lOO = 10 lbs. Ans, 

Example. — At what distance from the fulcrum on the long arm of 
a lever must I place a pound weight, to equipoise or weigh 20 lbs., 
suspended 2 inches from the fulcrum at the other end ? 
1 : 2 :: 20 : 40 inches. Ans, 

NoTK. — If weexaniino the foregoinff with reference to ihc fact, we shall see thai ererjr 
proposition In aimple proportion consisla of a tmn and a half! or, in other words, of a 
cumpound term coi)Hi»iJiig of two fiiciors, ami a factor for which another factor i« 9«^ii^hl 
that lotjelher sliall erjiial the rompujn*!. Wo have only to multiply the factors of the 
C(>in|x)un(l together — and a little oltscrvation will enahle ua to distinguish it — and divide 
by the renjaining factor, and the work ia accoinpliahed. See Compound Proportion. 



COMPOUND PUOPORTION, OR DOUBLE RULE OF THREE. 

Compound Proportion, like single proportion, consists of threr 

terms p:ivcn by which to find a fourtli — a divisor and two factors for 
formint? a dividend — but unlike singh3 proportion, one or more of the 
terms is a compoimd, or consists of two or more factors; and some- 
times a portion of the fourth term is given, wliich, however, is always 
a part of the divisor. 

Of the given terms, two are suppositive, dissimilar in their natures, 
and relate to each other, and to each other only ; and upon their rela- 
tion the whole is made to depend ; the remaining term is of the nature 
of one of the former, and relates to the fourth term, which is of tho 
nature of the other. 

The object sounbt is a number, which, multiplied into the factor or 
factors of the fourth term given, if any, and if not, which of itself, 
bears the same proportion to the dissimilar term to which it relates, 
as the suppositive term of like nature boars to the term to w^hich it 
relates. 

RrLE. — Observe the denomination in which the demand is made, 
and of tlie suppositive terms make that of like nature the secOnd, and 
the other the fust ; ma^e the remaining term the third term ; and, if 



I 



COMPOTTPiD PROPORTION. Ill 

there are any factors pertaining to the fourth term, affix them to the 
first ; multiply the second and third terms together and divide by the 
first, and the quotient is the answer, term, or portion of a term, 
sought. 

Example. — If 12 horses in 6 days consume 36 bushels of oats, 
how many bushels will suffice 21 horses 7 days^ 
12 X 6 : 36 :: 21 X 7 : J^. 

3 

30 X 21 X 7 147 

rr;; -z. — = = 73i bushels. An$, 

Ig X 2 ^ 

2 

Example. — If 12 horses in 6 days consume 36 bushels of oat«, 
how many horses will consume 73i bushels in 7 days ? 

36 : 12 X 6 :: 73i : 7 X •^• 
12 X 6 X 73i 147 
36 X 7 = -7- = ^^ ^°^^^^- ^^^- 

Example. — If the interest on $1 is 1.4 cts. for 73 days, (exact 
interest at 7 per cent.,) what will be the interest on $150.42 for 146 
days ? 

73 : 1.4 :: 150.42 X 116 \x. 

1.4X1.^^0.42X116 .^^, . 
-■ = ::?4.21. Ans. 

JO 

Example. — If the interest on SI is 1.2 cts. for 73 days, (exact 
interest at 6 per cent.,) what will be the interest on $125 for 90 
days? 

73 : 1.2 :: 125 X 90 : ^ = $1.85. Ans. 

Example. — If $100 at 7 per cent, gain $1.75 in 3 months, how 
much at 6 per cent, will $170 gain in II4 months? 

100 X "^ X 3 : 1.75 :: 170 x 6 x 11.5 : x, 
1.75 X 170 X 6 X 11-5 -T- 100 X 7 X 3 = $9.77,5. Ans. 

Example. — By working- 10 hours a day 6 men laid 22 rods of wall 
in 3 days ; how many men at that rate, who work but 9 hours a day, 
will lay 40 rods of wall in 8 days ? 

22 : 6 X 3 X 10 :: 40 : 9 X 8 X <:r. 

6X3X10X40-r-22X9X8 = 4y\-. Ans. 

Example. — If it costs $112 to keep 16 horses 30 days, and it 
costs as much to keep 2 horses as it costs to keep 5 oxen, how much 
will it cost to keep 28 oxen 36 days? 



112 CONJODiED PROPORTION, OR CHAIN RULE. 

16 X 30 : 112 : : f X 28 X 26 : x. 

Or,— 16 X 30 X 5 : 112 : : 28 X 36 X 2 Z x. 

nl 28 3I ^ _ 28 X 12 X_^ _ ^^^ ^3 ^^ 
10 30 5 5X5 

5 

Example. — If 24 men, in 8 days of 10 hours each, can dig a 
trench 250 feet long, 8 feet wide, and 4 feet deep, how many men, in 
12 days of eifrht hours each, will be required to dig a trench 80 feet 
long, feet wide, and 4 feet deep ? 
250X8X4 : 24 X8X 10 ::80X OX i : 12X8X^^=5— . Arts. 

Example. — If 120 men in six months perform a given task, work- 
ing 10 hours a day, how many men will be required to accomplish a 
like task in 5 months, working 9 hours a day? 

120 X C X 10 = 5 X 9 X X. 
Or, — 1 : 120 X X 10 :: 1 : 5 X 9 X "T. = 160. Ans, 

Example. — The weight of a bar of wrought iron, 1 foot in length, 
1 inch in breadth, and 1 inch thick, being 3.38 lbs., (and it is so,) 
what will be the weight of that bar whose length is 12.i feet, breadth 
3i inches, and thickness } of an inch? 

1 : 3.38 :: 12.5 X 3.25 X .75 : X, 
Or, — 1 : 3.38 :: V X V- X ? • '» and 

3,382^XilXi = 102.08+ Ihs. Ans, 
2X4X4 

Example. — The weight of a bar of wrought iron, one foot in length 
and 1 inch square, being 3.38 lbs., what length shall I cut from a bar 
whose breadth is 2^ inches, and thickness 4 inch, in order to obtain 
10 lbs.? 3.38 : 1 :: 10 : j^l x i X -^r. 

1 X 10 X 1 X 2 

= 2 feet l^rr inches. Ans. 

3.38 X 11 X 1 



CONJOINED PROPORTION, OR CHAIN RULE. 

The Chain Rule is a process for determining the value of a given 
quantity in one denomination of value, in some other given denomi- 
nation of value ; or the immediate relationship which exi^s between 
two denominations of value, by means of a cliain of approximate steps, 



CONJOINED PROPORTION, OR CHAIN RULE. 113 

circumstances, or equivalent values, known to exist, which connect 
them. In every instance at least five terms or values are employed 
in the process, and in all instances the number employed will be un- 
even. A proposition involving but three terms, of this nature, is a 
question in single proportion. The equivalent values employed are 
divided into antecedents and consequents^ or causes and effects ; and the 
value or quantity for which an equivalent is sought, is called the odd 
term. 

Rule. — 1. When the value in the denomination of the first antece- 
dent is sought of a given quantity in the denomination of the last conse- 
quent. — Multiply all the antecedents and the odd term together for a 
dividend, and all the consequents together for a divisor; the quotient 
will be the answer or equivalent sought. 

Rule. — 2. When the value in the denomination of the last consequent 
is sought of a given quantity in the denomination of the first antecedent. 
— Multiply all the consequents and the odd term together for a divi- 
dend, and all the antecedents together for a divisor ; the quotient will 
be the answer required. 

Example. — I am required to give the value, in Federal monoy, of 
5 Canada shillings, and know no immediate connection or relationship 
between the two currencies — that of Canada and that of the United 
States. The nearest that I do know is that 20 Canada shillings have 
a value equal to 32 New York shillings, and that 12 New York shil- 
lings equal in value 9 New England shillings, and that 15 New Eng- 
land shillings equal $2.50 ; and with this knowledge will seek the 
value, in Federal money, of the 5 Canada shillings. 
2^X1X32X5^ 
15 X 12 X 20 

Example, — If $2.J equal 15 New England shillings, and nine shil- 
lings in New England equal 12 shillings in New York, and 32 shil- 
lings in New York equal 20 shillings in Canada, how many shillings 
in Canada will equal $1 ? 

3 $ 

15 n ^0 1 

^1 g g^ = -¥■ = 5 shillings. Ans. 

3 i 

Example. — If 14 bushels of w^heat weigh as much as 15 bushels 
of fine salt, and 10 bushels of fine salt as much as 7 bushels of coarse, 
and 7 bushels of coarse salt as much as 4 bushels of sand, how many 
bushels of sand will weigh as much as 40 bushels of wheat? 

15 X 7 X 4 X 40 ,^1 u u 1 A 

— — — — =174- bushels. Ans. 

14 X 10 X 7 ^ 

10* 



114 PERCENTAGE. 



PERCENTAGE. 

Pure percentage, or percentage, is a rate by the hundred of a 
part of a quantity or number denominated the principal, or basis. 
But percentage, considered as a means, and as commonly applied, 
is mixed and related in an eminent degree ; and in this light may 
be regarded as divided into orders bearing dilTerent names. 

Thus Interest is percentage related to intervals of time in the 
past. 

Discount is percentage related to interest, and intervals of time 
in the future. 

Profit and Loss is comparative percentage, or percentage related 
to the positive and negative interests in business, etc., etc. 

Pure percentage is commonly called buokkr.\ge when paid to 
a broker for services in his line. 

It is called commission ^vhen paid to or received by a factor 
or commission merchant for buying or selling goods. 

It is called premium by an insurance company, when taken for 
insuring against loss. 

It is called primage when it is a charge in addition to the 
freight of a vessel, etc. 

Comparative percentage relates to the differences of quantities, 
and is confin(^d always to the idea of more or less. It implies ratio. 
This descrii)tion of percentage, though much in practice, seems not 
to be well understooil ; and often a (piantity is indirectly stated to 
be many times less than nothing, or many times greater than it is. 
The diilcrence of two quantities cannot be as great as a hundred 
per cent, of the greater, however widely unccpial the quantities 
may be, nor as small as no per cent, of the greater or lesser, how- 
ever nearly equal they may be. No quantity or number can be as 
small as 1 time less than another quantity or number; and there- 
fore cannot be as small as 100 per cent. less. But, since one quan- 
tity may be many by 1 time, or many times gi'cater than another 
with which it is compared, it may be said to be many by 100 times, 
or many hundred per cent, greater. 

When one of two quantities in comparison is stated to be three 
times less, or three hundred per cent, less, for instance, than the 
other, the expression is incorrect and absurd. The mi'aning evi- 
dently is, that it is two-thirds less, or only one-third as large as the 
other, — that it is QQ>r^ per cent, less, or only 33^ per cent, as large 
as the other. In common comparison, 1 is the measuring unit. In 
percentage, 100 is the measuring unit. 



PERCENTAGE. 115 

Let a "=. principal. 
h zz: percentage. 

s HZ amount (sum of the principal and percentage). 
d :=: difference of the principal and percentage. 
r = rate of the percentage. 
p HZ rate per cent, of the percentage. 

azizS'^bzizb'^rzHlOOb-^pznlOOs-^ (100 -|-J5), 

b z=is — azzzarzzzap -^ 100, 

p =z lOOr 1=^1005 -4- a zn 100(5 — a)-r'a, 

r Hzp -^ 100 izzb^azn (s — a) -^a^ 

s=za + b = a(l-{-r)z=z a(l 00 -]-p) -^ 100, 

dzHa — bz=:2a — sins — 2b hz a(l — r). 

Tojind the Percentage* 

EXAMPLES. 

What is I of 1 per cent, of $200 ? 

^ =: ar = a;) •— 100 =: $0.50. Ans. 
1 of 2 per cent, of 50 is what part of 50 ? 

7X100 ='^' ^'^• 
What is f of I of ^ of 24 per cent, of 150 lbs. ? 
150 X 12 -^ 100 = 18 lbs. Arts. 
What is 2| percent, of 19 bushels ? 

J_9 X -jW^ 0.45125 bushels. Ans. 

Bought a job lot of merchandise for $850, and sold it the same 
day, brokerage, 2-^ per cent., for $975 ; what was the net gain? 
5 — sr — a=is — (sr -{- a) r= 5(1 — r) — azzi 975 — 975 X .025 

— 850 = $100,625, Ans. 

To find the Rate or Rate Per Cent 

EXAMPLES. 

What per cent, of $20 is $2 ? 

?^zi:&~a, j9iz: 100&-^a= 10 per cent. Ans. 

12 dozen is equal to what per cent, of 2 dozen ? 
12-^2 = 6, GOO per cent. Ans. 



116 PERCENTAGE. 

What part of 5^ lbs. is f of 2 lbs. ? 

r=^X i^ = ii = 0.27-^\. Afis. 

24 J ^er cent, is what per cent, of 36 3 per cent. ? 

CG§ per cent. Ajis. 

For an article that cost S4, S5 were received ; what per cent 
of S4 was received ? 

7? zz: 5 X 100 -^ 4 — 125 per cent. Ans. 

A farmer sowed 4 bushels of wheat, which produced 48 bn>hcls; 
what per cent, was the increaae ? 48 is jnnrc than 4 by what per 
cent, of 4 ? The difference of 48 and 4 is what per cent, of 4 ? 

b ' p 4 

100(48 — 4)-^ 4 = 1100 percent, Ans. 

What per cent, would have been the tlccrcase, if ho had sowed 
48 bushels, and harvested only 4 bushels? 4 is less than 48 by 
what rate of 48 ? The dillerence of 48 and 4 is what per cent, of 

48 V 

rz=i{a — h)-^az=z\ — =0.01 J, or 91 § percent. Ans, 

Since water is composed of 8 atoms of oxygen and 1 atom of 
hydrogen, what per cent, of it is oxygen V 8 is what per cent, 
of the sum of 8 and 1 ? 

« , ^^ 100a 8 ^^„^ 

a-\'h a-YO ' a-f-6 8 -}- 1 

or 88.89 - per cent. Ans. 

What per cent, of it is hydrogen ? 1 is what per cent, of the 

sum of 8 and 1 ? 

a h 100ft 1 

r z= 1 j— r ■=! — , -J , p — — r-z = — ,-- zn . 1 ] ] 1 -1- or 

a-\- h a-\-h n-\-b 8-(-l ' 

11.11 -|- per cent. Ans, 

How many volumes of water must be added to 100 volumes of 
90 per cent, alcohol to reduce it to 50 per cent, alcohol or common 
proof? 90 is more than 50 by what per cent, of 50 ? The diller- 
ence of 90 and 50 is what per cent, of 50 ? 

(a — h)\00 (90 — 50)100 ^^ . 
P = - F— = - 50 ^^^' '^'^' 



I 



PERCENTAGE. 117 

How many volumes of 50 per cent, alcohol must be added to 
100 volumes of 90 per cent, alcohol to produce 80 per cent, alcohol ? 
90 is more than 80 by what per 4*cut. of the difference of 80 and 
60 ? The diff*ercnce of 90 and 80 is what per cent, of the differ- 
ence of 80 and 50? 

(a — &)100 (90 — 80)100 ^^, 

How many volumes of 90 per cent, alcohol must be added to 100 
volumes of 50 per cent, alcohol to raise it to 80 per cent, alcohol ? 
50 is less than 80 by what per cent, of the difference of 90 and 80 ? 
The diffc'Tence of 80 and 50 is what per cent, of the difference of 
90 and 80 ? 

n,^hf)lOO (80 — 50)100 „^^ 

a — 90 — 80 

If to 2 volumes of 95 per cent, alcohol, 1 volume of 50 per cent, 
alcohol be added, what per cent, alcohol wmU be the mixture? The 
sum of 50 and twice 95 is what per cent, of the sum of 2 and 1 ? 

2a + b 2 X 95 + 50 ^^ 

-^- =--^-p^— = 80percent. Ans. 

In a barrel of apples, the number of sound ones was 60 per 
cent, greater than the number that were damajied. What per 
cent, less was the number that were damaged than the number 
that were sound ? CO per cent, is what per cent, of the sum of 
100 per cent, and 60 per cent. ? .G is what rate of 1 -|- .6 ? 

100 O.a 1 60 

3 7-^ per cent. Ans. 

Since the number of damaged apples was 37^ per cent, less than 
the number that were sound, what per cent, greater was the num- 
ber that were sound than the number that were damaged ? 

r == a + (1 — a) = 1 + (1 — a) — 1 = 60 per cent. Ans. 

Since the number of sound ones was 60 per cent, greater than 
the number that were damaged, what per cent, of the whole were 
sound ? 

a + a2 I -I., a 100 + 60 ^^ 

- — — ;;= -| =80 per cent. Ans. 



2a 2 ' ^ 2 

What per cent, of the whole were damaged ? 

(100 — 60) + 2 = 20 per cent. Atis 



118 PEKCENTAGE. 

Since 20 per cent, of the apples were damaged, what per cent, 
less was the number that were damaged than the number that were 
sound ? 

1 — 2. a . 1 100 — 2a ,^^ 100 



— 1 . p= =100 



2 — 2. a 2 — 2.a^ 200 — 2a 200 — 40 

37^ per cent. Ans. 

What per cent, greater was the number that were sound than 
the number that were damaged ? 

r=2 — (1 -|-2.a) = 2 — 2. a — 1 = 60 percent. Ans. 

Since 80 per cent, of the whole were sound, what per cent, less 
was the number that were damaged than the number that were 
sound V 

2.a— 1 ^ 1 2X.«0 — 1 ^^, , . 

r:= = 1 = ^^ = 37i per cent. Ans. 

2. a 2. a 2 X.80 * 

Since the number of damaged ones was 3 7^ per cent, loss than 
the number that were sound, what per cent, of the whole were 
sound? 

1 100 100 • ^ ^ ^ 

^ = 2-^27^- ^^2::r2^= 2-2X87.5 = '^P^^^^"^- ^"^- 

Since 80 per cent, of the whole were sound, what per cent 
greater was the number that were sound than the number that 
were damaged ? 

2 a • 

r= ^=2.a — 1 = 2 X .80 — 1 = GO percent. Ans. 

2 

Lost 20 per cent, of a car^o of coal by jettison, and 5 per cent, 
of the remainder by screenmg, what per cent, of the coal was 
saved ? 

' = (1 — r') (1 — r") = (1 — .20) — (1 — .20) 
.05=i(l— .20)(1— .05)=76 percent. Ans. 



df--b'f = dff^ X . 
d"'-b"' = d"'y &c. 



Yesterday drew 12 per cent, of my balance of S4,273 in the 
bank, and deposited Si, 000 ; and to-day have drawn 31^ per cent, 
of the balance left over, or as it stood last night. What per cent, of 
the sum of the first-mentioned balance and deposit of yesterday 
have I drawn ? 

6' + &'' 512 + 1487.575 o-,no.. . . a 

r = ^— 1 = J — . = 37.9354 + per cent. Ans. 

a + 771 4273+1000 ^ ^ 



PERCENTAGE. 119 

What per cent, of the said sum is remaining in the bank ? 

1 -. — i = i — r= 62.0646 — 

a-f-m a-\-m a-^-m j, a 

•^ ' ' per cent. Ans. 

What per cent., predicating it upon the first-mentioned balance, 
have I drawn ? 

h' + h" 512.76 + 1487.576 .a^^oA ^ a 

rzz:_-II — z=z ' 1=46.8134- per cent. Ans, 

a 4273 

What per cent, have I drawn, predicating it upon what I now 
have in the bank ? 

percent. Ans. 
What amount of money must I deposit to make good 62^ per 
cent, of the aforementioned sum ? 

J = r(a + wi) + 5' + 5'' — (a + 77i)=zr(a + 7n) — fZ'' = 

S22.96. Ans. 

To find the Principal or Basis, 

EXAMPLES. 

The percentage being 250, and the rate .06, what is the 
principal ? 

a=:5-T-r=:100&~p=250-i-.06 = 25,000 + 6=4,166f. Ans, 

A tax at the rate of | of 1 per cent, on the valuation was S27.50. 
What was the valuation ? 

&X 6X100 ^„„^^ 

a=~- f^ = $3,300. Ans, 

5 

Sold 120 barrels of flour, which amounted to 12 per cent, of a 
certain consignment. The consignment consisted of how many 
barrels ? 

120 + 0.12 = 1,000. Ans. 

216 bushels is more by 8 per cent., or 8 per cent, more, than what 
number of bushels ? 8 per cent, more than what number is equal 
to 216 ? What number, plus 8 per cent, of it, will make 216 ? 

a = 5 + (1 + r) = 216 + 1.08 = 200. Ans. 

200 lbs. is less by 8 per cent., or 8 per cent, less, than what num- 



120 INTEREST. 

ber of lbs. ? 8 per cent, less than what number is 200 ? What 
number, minus 8 per cent, of it, is equal to 200 ? 

a = d-^ (I— r) = 200-^(1— .08) = 2 1 7^^. A ns. 

. • . 2l7^\—2n^\ X.08 = 200=a — 6 = f/ = a(l — r). 

To a quantity of silver, a quantity of copper equal to 20 per 
cent, of the sliver is to be added, and the mass is to weigh 22 
ounces. What weight of silver is required ? 

a=iS^(l-\- r) =3 22 -^ 1.2 — 18 J ounces. Ans. 

What weight of copper is required ? 

s sr 

s — 7-, =—:— = 3* ounces. Ans. 
I -\-r l-}-r * 

To a quantity of copper, a (quantity of nickel ecpial to 62} per 
cent, of the copper, a (juantity of zinc e(jual to .'5.3^ per cent, of the 
copper, and a ipiantity of lead ecjual to 5 per cent, of the copper, 
are to be added; and the whole is to weii^h 10 J jx)unds. The 
weight of each constituent of the alloy is required. 

_ s 40^ 

^"~ 1-f r + r' + W'""l-f-.62i-f.33j+.05 

= 20 lbs. of copper, 
fc = 20 r= 12^ lbs. of nickel, . 
hf= 20 W = G § lbs. of zinc, ^ ^^' 
h'f = 20r''=l lb. of lead. 



INTEREST. 

Universal for any rate j^er cent. 

T = time in months and decimal parts of a month ; /= time in days ; 
P = principal ; r = rate per cent., expressed decimally; t= interest. 

PXTXr PX^Xr 






12 365 



I2j_365t rp_12j 365 i __12t_365t 

Example. — A promissory note, made April 27, 1864, for 



INTEREST. 121 

$8251*/^ and Interest at 6 per cent., matured Oct. 6, 1865 : what 
was the interest ? 



Oct. is lOth month. 
April is 4th month. 

F. m. d. 

1865 . 10 . 6 

'64 . 4 . 27 



Time from April 27 to Oct. 6 (one of 

the dates always included) r= 162 days, 

which, added to the 365 days in the year 

preceding = 527 days. 

Note. — One day's interest at least is gener- 
ally lost by computing the time in years and 
months, or months, instead of days. 



Time= 1.5.9 

825.25 X 17.3 X .06 -^ 12 = $71.38. Am. 
825.25 X 527 X .06 -^ 365 = S71.49. Ans. 

To find a constant divisor^ ^ifor any given rate 2^er cent. 

When the time is taken in months, ^' = 1 2 ~ r. 

When the time is taken in days, k = 365 -— r ; thus, 

p X t 
When the rate is 6 per cent. -^Y)^oq~= Interest. 

P X < 
When the rate is 7 per cent. .^^. = Interest, &c. 

Example. — Required the interest on $750 for 93 days, at 7 
per cent. 

750 X 93 -^ 5214 = $13.38. Ans. 

Example. — What is the rate per cent, when $450 gains $94^ 
in 3 years? 

450 : 100 :: 94.5 : 3.^= 7 per cent. Ans. 

94.5 -^ 3 X 450 =r .07. Ans. 

Example. — In what time will $125 at 6 per cent, gain $18|? 

6 : 100 :: I8.75 : 125 X ^• = 2^ years. Ans. 

18.75 — 125 X.06 = 2^ years. Aris. 

Example. — What principal at 5 per cent, interest will gain 
$16| in 18 months? 

5 : 100 :: I6.875 : 1.5 X a: = $225. Ans. 

16.875 X 12 -^ 18 X .05 = $225. Ans. 
11 



122 COMPOUND INTEREST. * 

When partial payments have been made. 

Rule. — Find the amount (sum of the principal and interest) 
up to the time of the first payment, and deduct the payment there- 
from ; then find the interest on the remainder up to the next pay- 
ment, add it to the remainder, or new principal, and from the sum 
subtract the next payment ; and so on for all the payments ; then 
find the amount up to the time of final payment for the final 
amount. 



COMPOUND INTEREST. 

If we calculate the interest on a debt for one year, and then on 

the same debt for another year, and a;zain on the same debt for 
still another year, the sum will be the simple interest on the debt 
for three years. But, on the contrary, if we calculate the interest 
on the debt for one year, and then on the amount (sum of the prin- 
cipal and interest^ for the next year, and then on the second 
amount for the tliird year, the sum of the interest so calculated 
will be the compound interest, or yearly com|)Ound interest, on the 
debt for three years; equal to the simple interest on the debt for 
three years, j)lus the yearly compound interest on the first year's 
interest for two years, ])lus the sim|)le interest on the second year's 
interest for one year. So, if we divide the time into shorter 
periods than a year, and proceed tor the interest as last sujrgested, 
the interest will be compound. Thus we have half-yearly com- 
pound interest, or compound interest semi-annually, quaiter- 
yearly compound interest, or compound interest quarterly, &e. 

This method of computing interest is predicated upon the 
natural idea, that interest, when it becomes due by stipulation and 
is withheld, commences to draw interest, and continues at use to 
the holder, at the same rate as the principal, until it is paid, like 
other over-due demands ; and that the interest so made matures 
and becomes due as often, and at the same periods, as that on the 
principal. 

It will be perceived by the foregoing that the xcorking-time in 
compound interest is the interval between the stipulated payments 
of the interest, or between one stipulated payment of the interest 
and that of another ; and that the icorking-rate is pro rata to the 
rate per annum. 

Thus the amount of $100 at semi-annual compound interest for 
2 years, at 6 per cent, per annum, is 



COMPOUND INTEREST. 123 

100 X (1.03)4 = $112.550881 =$112.55, or 
100. 
.03 

3. 

100, 

103, 

.03 , 

3.09 5 

103. 



106.09 
,03 

3.1827 
106.09 

109.2727 
j03 

3.278181 
109.2727 

Sll 2.550881, as before. 

if we let P = principal or debt at interest, 
r = working-rate of interest, 

n = number of intervals into which the whole time is 
divided for the payment of interest, or number of consecutive 
intervals for the payment of interest that have transpired without 
a payment having been made, 

i = compound interest, 
A = P -4- ^* or amount, then 

^=(l+r)-t = A-P. 

Example. — What is the compound interest, or yearly com- 
pound interest, on $100 for 1^ years, at 6 per cent, a year ? 

lOOX 1.06 X 1-03 = 109.18 — 100 = $9.18. Ans. 

Example. — What is the amount of $560.46, at 7 percent 
compound interest per year, for 6 years and 57 days ? 

5^0.46 X (1.07)6 x(l + '^3y^ ) = $850.29. Ans. 



liA COMPOUND INTEREST. 

Example. — The principal is S250, the rate 8 per cent, a year, 
the time 2 years, and the interest compound per quarter year: 
required the amount. 

250 X (l. —Y =$292.91. Ans. 

When Partial Payments have been made. 

Rule. — Find the amount up to the fn-st payment, and dedact, 
the payment therefrom ; then find the amount up to the next pay- 
ment, and therefrom deduct tliat payment ; and so on for all the 
payments; then find the amount up to the time of final payment, 
for the final amount. 

Example. — A note of hand for S500 and interest from date, 
at G per cent, a year, has been paid in part as follows ; viz., two 
years and four months from the date of the note, by an indorse- 
ment of S50 ; and three years from that indorsement, by an in- 
dorsement of SI 50. It is now eight months since the last payment 
was made, and the demand is to be settled in full: re(|uin.'d the 
amount at the present time, interest being compound per year. 

500 X (1.06)« X 1.02 — 50 = 523.036 

(1.0 6)» 

622.944 
150 

472.944 
1.04 



S491.86. Ans, 

The following table shows (l-\-r) raised to all the integer 
powers from 1 to 12 inclusive ; r being taken at 4, 5, 6, 7, 8, and 10 
per cent. If the numbers in the column headed years are taken 
to represent years, then 4 per cent., 5 per cent., &c., at the head 
of the columns of powers, will stand for per cent, per annum : if 
they are taken to represent half-years, then 4 percent., 5 percent., 
&c., will stand for per cent, per half-year, &c. The quantities in 
the columns arc powers of (1 +r), of which the numbers referred 
to and standing opposite, respectively, arc the exponents. Thus, 
1.26248, in the 6 per cent, column, and against 4 in the column 
marked years, = (1.06)* ; and so with the others. The powers 
or quantities in the columns are co-efflcients in the calculations. 



COMPOUND INTEREST. 



125 



Years. 


4 per cent. 


5 per cent. 


6 per cent. 


7 per cent. 


S per cent. 10 percent. 


1 


1.04 


1.05 


1.06 


1.07 


1.08 


1.10 


2 


1.0816 


1.1025 


1.1236 


1.1449 


1.1664 


1.21 


3 


1.12486 


1.15762 


1.19102 


1.22504 


1.25971 


1.331* 


4 


1.16986 


1.21551 


1,26248 


1.3108 


1.36049 


1.4641 


5 


1.21665 


1.27628 


1.33823 


1.40255 


1.46933 


1.61051 


6 


1.26532 


1.3401 


1.41852 


1.50073 


1.58687 


1.77156 


7 


1.31593 


1.4071 


1.50363 


1.60578 


1.71382 


1.94872 


8 


1.36857 


1.47746 


1.59385 


1.71819 


1.85093 


2.14359 


9_ 


1.42331 


1.55133 


1.68948 


1.83846 


1.999 


2.35795 


10 


1.48024 


1.62889 


1.79085 


1.96715 


2.15892 


2.59374 


11 


1.53945 


1.71034 


1.8983 


2.10485 


2.33164 1 2.85312 


12 


1.60103 


1.79586 


2.0122 


2.25219 


2.51817 1 3.13843 



Note. — If a co-efficient is wanted for a greater number of years or intervals 
of time than is given in the table, square the tabuhir co-efficient opposite half 
that number of intervals, or cube the tabular co-efficient opposite one-third 
that number of intervals, &c., for the co-efficient required. Thus, 

1.9992=1,586873 = 1.0812 X 1.086=1.0818 = 3.996, 

the co-efficient for 18 years or intervals at 8 per cent, per interval, &c. 

If the compound interest alone is sdught on a given principal, subtract 1 
from the tabular power corresponding to the time and rate, and multiply the 
remainder by the given principal ; the product will be the compound interest. 
Thus (1.26532--1) X 100 = $20,532, the yearly compound interest, at 4 per 
cent, per annum, on $100 for 6 years, or the half-yearly compound Interest, at 
8 per cent, per annum, on $100 for 3 years, or the half-yearly compound inter- 
est, at 4 per cent, per half year, on $100 for C half-years. 

Example. — What is the amount of S125.54, at 5 per cent, 
compound interest, for 7 years, 21 days ? 

21 X 05 
^"1 Qoi — = 1.00288, the co-efficient for the odd days; and, 

turning to the 5 per cent, column in the table, we find against 7, in 
the column of years, 1 .4071, the co-efficient for 7 years : then 

125.54X1.4071X1.00288 = 81 78.20. Ans. 

Example. — In what time, at 7 per cent, compound interest 
per annum, will SIOOO gain $462? A-^-P = (1 + r)» : then 
1462 -f- 1000 zi: 1.462, the co-efficient demanded. Turning now 
to the 7 per cent column in the table, we find the nearest less 
co-efficient there (there being none that exactly corresponds) to 

/ 1.462 \ 

be that for 5 years ; viz., 1.40255. And \7~77^^^ — 1) -i-.07 = 

.60553, the fraction of a year over 5 years to the answer. 

.60553 X 365 = 221 days; 5 years, 221 days. Ans. 
11* 



126 



COMPOUND TXTEREST. 



The following table is of the same nature as the preceding, 
and is applicable when the interest becomes due at regular inter- 
vals short of a year, or when the working-rate in compound inter- 
est is less than 4 per cent. 

The quantities in the 1 J per cent, column apply to quartei^y early . 
compound interest when the rate is 7 jkt cent, a year ; and those 
in the 1^ per cent, column, to quarterly comi>ound interest when 
the rate is 5 per cent, a year ; also tlie form<»r are applicable to 
monthly compound interest at 21 per cent, per annum, and the 
latter to monthly compound interest at 15 jxjr cent, per annum; 
and so relatively, throughout the table. 



.♦ 


I 


1 


a 
3 

*• 


1 


§ i 5 


*: 


1 




i 




E 

00 


& 
S 


i 




i 


& 




1 


1.035 


1.03 


1.025 1.02 


1.0176 11.015 


1.0125 ll.Ol 


1.006 


2 


1.07128 


1.0609 1.050631.0404 


1.0353! I.o.?r>o3 i.ft2510'l ooftt 


T. 01003 


8 


1.10872 


1.002731.07689 1.0612111.0-,:^; ' " 


- 1<? 


4 


1.14752 


1.12551 1.10381 1.08243, 1.071 


' ", 


6 


1.187GD 


1.15927 1.13141 1.10408|l.O'.M)<,^ :. ... .. . 


... - -5 


6 


1.22925 


1.19405 1.15969 1.1261G|1.1077 ll.«nW44|1.0774 1.06152 


i.()au38 


7 1 1.27228 


1.22987:1.18869 1.14869|1. 12709 1.109S4 1.090h7 l.<>7lM1 


i.or,f.3 


8 


1.31681 


1.26677 1.2184 !1. 1716611. 14681'!. 12649 1.10451 1." 


"1 


9 


1.3629 


1.30477 1.24886 1.19509il.l6688;1.14.3;J9 1.118311.' 


I 


10 


1.4106 


1.34392 1.2800- t.-- ij873 Il.l6(»rj4ll. 13229 1.1 




11 


1.45997 


1.3S423 1.31L' 1.20S0S 1.17795|l. 14045 1.1 i 




12 1.61107'1.4257« 1.344- I J 1 1.22J»22 1. 195G2'l. 100781.1- 


■ ''_li 



Example. — ^Vhat is the amount of S750 for 4 years and 40 
days, allowing half-yearly compound interest, at 7 [)er cent, a year? 

In this case, the working-rate tor the full periods of time is 8^ per 
cent, and there arc 8 such full periods ; then, sei'king thtj eo-eflicient 
in the 3J per cent, column, we find against 8, in the column of 

times, the quantity or co-efficient 1.31681 ; and 1 + — ^~ — = 

1.00767: therefore 

750 X 1.81681 X 1.00767 = 8995.18. Ans. 

Example. — AVhat is the amount of SIOOO at compound inter- 
est per quarter-year, at H per cent, per (piarter-year, lor 4^ years? 

1000 X 1.126492 X 1.015 = $1288.01. Ans. 



DANK INTEREST. 127 



BANK INTEREST OR BANK DISCOUNT. 

A bank loans money on a promissory note made payable with- 
out interest at a future period. The operation is called discounting 
the note at bank, and is as follows : The bank takes the note, finds 
the interest on it for three days more time than by its own tenor it 
has to run, subtracts it from the principal, and hands the balance, 
called the avails of the note, in its own bills, to the party soliciting 
the loan, or otfcring the note for discount, as it is called ; whereby 
the note becomes the property of the bank, and the maker and 
endorsers are held for its payment when it matures. 

The three days mentioned arc called days of grace^ and the 
note does not become due to the bank until three days after it 
becomes due by its own tenor. These proceedings are sanctioned 
by usage, and protected by law. 

Bank interest, then, is bank discount, and bank discount is bank 
interest. But bank discount is not discount^ nor is it what is called 
legal interest on the money loaned. It is the interest on the money 
loaned, plus the interest on the interest of the loan, plus the inter- 
est on the difference of the sum taken and the interest on the loan 
for the time of the loan ! A kind of interest more onerous, if any 
description of interest be onerous, than comjwund interest, rate 
for rate and time for time, as may be readily perceived. 

Let P = principal or face of the note. 

r = working-rate of the interest for the time of the loan. 
a ^= avails of the note or sum borrowed. 
i = bank interest. 
i = time of the loan. 

R : r : : T : ^ R being the rate per cent, per annum, and T 
one year. 

P=ra-^(1— r). arrrP — Pr. { = Pr. r z= (P — a) -f- P. 

If we let n represent the time of the note in months, 
Rn , 3R 

^' — Y^ ~T" o^* But it is the practice with many banks to count 

the days of grace as so many 360ths of a year. 

Putting d to represent the time of the note in days, 

r=z 31 true time and rate. 

365 ' 

With some banks, it is the practice, in calculating interest, to take 
the time, when it does not exceed 93 days, as so many 360ths o^a 
year. 

A note having 3 months to run from Aug. 10, for instance, will 



128 



BANK INTEREST. 



fall due Nov. 10-13; but one having 90 days to run from Aug. 
10 will fall Nov. 8-11. The time including grace of the former 
is 3 mo. 3 ds., and that of the latter 3 mo. 2 ds., mean lime. Never- 
theless, the former embraces 95 days, or one day more than mean 
time, and the latter but 93 days. 

The following table shows 1 — r, mean time, for the intervals of 
time set down in the left-hand column ; K being taken at 4, 5, 6, 7^ 
and 8 per cent, per annum, as set down at the top of the columns. 



Time. 




4 


6 


6 


7 


8 


mo. 


dt. 


per cent 


percent. 


percent. 


per cent 


p«r Mmt. 


1 


3 


.90^333 


.995417 


.9945 


.993583 


.992667 


2 


3 


,993 


.99125 


.9895 


.98775 


.986 


3 


3 


.989007 


.987083 


.9845 


.981917 


.979333 


4 


3 


.980333 


.982917 


.9795 


.976083 


.972667 


6 


3 


.983 


.97875 


.974.') 


.97025 


.966 


6 


3 


.979007 


.974583 


.9095 


.964417 


.959333 


7 


3 


.970333 


.970417 


.9045 


.958583 


.952667 


8 


3 


.973 


.96025 


.9595 


.95275 


.946 


9 


3 


.909007 


.902083 


.9545 


.946917 


.939333 


10 


3 


.900333 


.957917 


.9495 


.941083 


.932667 


11 


3 


.903 


.95375 


.9445 


.93525 


.926 


12 


3 


.959667 


.949583 


.9395 


.929417 


.919333 



Putting k to represent the tabular quantity 1 — r, 

a=Vlc, r- a~ A:, i = P — a = P— Pt. 

Example. — What will l)e the avails of a note for 81,250 
payable in 4 months if di^'ountetl at a bank, interest being 7 per 
cent, a year ? 

The tabular constant 1 — r, in the 7 per cent, column, against 4 
months and 3 days in the time column, is .976083, and 
Sl,250 X .9760831=81,220.10. Ans. 

Example. — For what sum must I make a note having 6 months 
to run, in order that the avails at bank, if discounted on the day 
of the date of the note, may amount to S956.38, interest being 6 
per cent, per annum ? 

By the table, S056.3S-^. 0095 zz: $986.4 7. Ans. 
Example. — What is the rate of bank interest when the nomi- 
nal or legal rate is 7 per cent. V 

.07 -^ (1 — .07) = .07527 = 7^ -f- ^ f^ per cent Atis. 

Note. — A note having 5 months to run from Feb. 1 will fall due July 
1-4; and the time, including grace, is 5 mo. 3 ds.= 155 days, mean time. 
But the time in days from Feb. 1 to July 4, when February has but 2i? days, 
is 163 days only, or 2 days short of mean time. 



DISCOUNT. — COMPOUND DISCOUNT. 129 



DISCOUNT. 

Discount is a deduction of the interest on the present worth or 
availability of a debt not yet due, in consideration of its present 
payment. The principal is the present nominal value of the debt, 
interest included,' if any interest has accrued. The time js the 
interval from the present to the date at which the debt will become 
due. The rate is the legal rate of interest, if no other rate is speci- 
fied ; and the present worth is that sum of money, which, if put at 
interest at the same rate and for the same time as the discount, will 
amount to the principal. 

Let a represent the principal, d the' discount, w the present 
worth, and i the interest on one dollar for the time and at the rate 
of the discount. 

w=:a-^(l -[- 1) zn a — d. d^=.ai-^ (l~h ^-* ^ — ^' 
azzid {l-{'i)-^izi=, d-\-w. 

Example. — Required the discount on S250 for 8 months at 6 
per cent. 

The interest on Si for 8 months at 6 per cent, is .04 of a dollar, 
or 4 cts. ; and 

250 X .04~(l+.04)z:=S9.6154. Ans. 

Example. — Required the present worth of Si 2 72.6 2 due 247 
days hence, discount 7 per cent. 

The interest on $1 for 247 days at 7 per cent. =: 247X.07 -^365 
= 0.04737, and 

1272.62-^- 1.04737 =81215.06. Ans. 

Note. — *' Talcing off^ in common parlance, a certain per centum from 
the face of a demand, is equal to deducting the interest, at that rate per cent- 
tum, on the present worth for 1 year, plus the interest on the interest of the 
present Worth, at the same rate per centum for 1 year. 



COMPOUND DISCOUNT. 

Compound Discount is to compound interest what simple dis- 
count is to simple interest. In both cases of discount, the differ- 
ence between the principal and the discount is that sum of money, 
which, if put at interest for the same length of time, at the same 
rate, and in the same general manner as the discount, will amount 
to the principal. 

Rule. — Add 1 to the rate per cent, of the discount for the 



130 COMPOUND DISCOUNT. 

worklng-tlme, and raise the sum to a power corresponding with tlio 
number of working-times; divide the principal by the power, and 
the quotient will be the present worth ; subtract the present worth 
from the principal, and the remainder will be the compound 
discount. 

Note. — The tables of the powers of 1 + r, applicable to compound In*^ 
terest^are ecjually applicable to comi)ouud discount. 

Example. — Required the present worth of a debt of S250, 
allowing yearly compound discount, at 7 per cent, a year, for 
3 years 84 days. 

07 y 84 

1 +* = 1.01011, the worklnfr-ratc for the 84 davs, and 

' • 365 '^ 

250-^(1.073 X 1.0ir,ll)=r 8200.84. Ans, 

Example. — What is the present wortli of a debt of $150.25, 
due 3 years, 3 months, and 10 days hence, without interest, allow 
ing compound discount per quarter-year, at 1 J per cent, per quar- 
ter-year V 

150.25 -^(^ 1.015" X 1.*^-- '^-^- W^;w. 
\ 365 / 

By table, 150.25 -f- (1.11)502 X 1.015 X 1.00164) = 

S123.61. Ans. 

Note. — What Ls lioro (Irnominntod the debt, or principal, represents the 
debt at the close of the time of the di.-'count ; that is, if the debt be on in- 
terest, the interest must be included in what is here culled the debt, or 
principal. 



PROFIT AND LOSS. 

The term *' Profit and Loss," as intimated in treating of 
Pkkcentage, relates to the positive and negative interests in 
business, and embraces the idea of both. 

Both profit and loss are absolute quantities, and are expressed by 
the diilerence of the cost price and selling price that limit them. 
They are usually, however, estimated by percentage, predicated 
upon the first-mentioned price or prime cost. 

When the selling price is greater than the cost price, or when 
the money obtained by the disposal of property exceeds what the 
property cost, the difference is positive, and denotes increase, 
profit, or gain. Conversely, when the cost price is greater than the 
selling price, or when property is disposed of for less money than 
it cost, the difference is negative, and denotes decrease, loss, or 



PROFIT AND LOSS. 131 

waste. So, the difference of the two prices, divided by the cost 
price, expresses the rate of gain on the cost when the selling price 
is the greater, — expresses the rate of loss on the cost when the 
cost price is the greater. 

Let c represent the cost price, purchase price, par value, or sum 
of money paid for the property ; 5, the selling price, trade price, 
premium price, or sum of money received in exchange for the 
property ; r, the rate of the profit or loss ; p, the rate per cent, of 
the profit or loss. 

To find the rate or rate j^er cent, of the profit or loss. 

z=z - "^ ^ . Moreover, when the difference is 



c 

s s 
positive, r = 1 ; and, when it is negative, r zz: 1 . 

Example. — Paid $4 for an article, and sold it for So. What 

per cent, was gained ? 5 is more than 4 by what per cent, of 4 ? 

The difference of 5 and 4 is what per cent, of 4 ? 5 — 4i=: $1, 

5^4 
gained; and — - — zzz.25 = |- — 1. 25 per cent. Ans. 

Example. — Paid $5 for an article, and sold it for S4. AVhat 
per cent, was lost ? 4 is less than 5 by what per cent, of 5 ? The 
difference of 4 and 5 is what per cent, of 5? 4 — b:=z — lziz$l, 

5 r-s^ 4 

lost ; and ^ ziz .20 = 1 — |. 20 per cent. Ans. 
o 

Example. — A whistle that cost 3 cents was sold for 20 cents I 
The profit was how much per cent ? (20 .-^ 3) -^ 3 ^: 5§ or 566§ 
per cent. Aiis. 

Example. — A fop paid SIO for a well-made and well-fitting 
pair of boots for his own wear, that were worth what they cost him ; 
but, being told that they were unfashionably large, sold them for 
$4. His vanity cost him what per cent, of the purchase price ? 
1 — y^^zz: .6 or 60 per cent. Aiis. 

To find a price long a given per cent, of the cost, or to find a sell- 
ing price that shall he the sum of the cost price and a given per 
cent, of it. 

5 zz: c + cr z= c (1 +?') = c (100 -\- p) ~ 100. 

Example. — At what price must I sell an article that cost 
$2.35 to gain 25 per cent. ? 2.35, more 25 per cent, of it, is how 
much? The sum of $2.35 and 25 per cent, of it is how much? 
2.35 -[- 2.35 X .25 = 2.35 X 1-25 :zz $2.93f . Ans. 



132 EQUATION OF PAYMENTS. 

To find a price short a given per cent, of the cost, or to find a sell- 
inrj price that shall he the difference of the cost price and a given 
per cent, of it. 

s=ic — crz=:c (I — r)=zc (100 — ;>)-^lOO. 

Example. — I have a damaged article of merchandise that cost 
$2.75, and I wish to mark it Tor sale at 30 per cent, below cost. 
At wliat price shall I mark it ? 2.75 less 30 per cent, of it is how 
much? Tlie diirerence of 82.75 and 30 per cent, of it is how 
much V 2.75 (1 — .30) = 2.75 X -7 = Sl.925. Ans. 

To find the cost price when the selling price and profit per cent, are 

given. 

s=^c-\-cr:=^c (1-f-r). •. c = .s -^ (1 -(-'') ^= luO .s- -i- (10<> -[-;;). 

Example. — What cost that article whose selling price, S4, is 
long 25 per cent, of the cost V What price, more 25 per cent of 
it, is e([ual to 84 ? 81 is the sum of what price and 25 per cent 
of it? 400-^125 = 83.20. Ans. 

To find the cost price when the selling price and loss per cent, are 

given. 
sz=c — cr:z=c(l — r) .*. rzzi .s -J- (1 — /) = inO .s-i- (100 — />) 

Example. — What cost that article whose selling price, 8375, 
is short 7 per cent, of the cost V What price loss 7 per cent, of it 
is ecpial to S3 75 ? S3 75 is the difference of what price and 7 per 
cent, of it ? 
375 -^- (1 — .07) = 375 -^ .93 — 375 X 100 -|- (100 — 7) = 

8403.226. Ans. 



EQUATION OF PAYMENTS, OR AVERAGE. 

Average consists in finding the time at which several sums, 
falling due at dillbrcnt dates, become due if taken collectively. 

Rule. — Multiply each sum respectively by the num]>er of 
days it falls due later than that falling due at the earliest date, and 
divide the sum of tlie products by the sum of the several sums. 
The (quotient will be the number of days subsecjuent to the earliest 
date at which the whole will mature, or averages due. 

N(1TE. — Average pivos no ** intcrrat on intere.'it " to the creclitor. It does 
not ^ive liim his just due. It estiiniitis by way of thi' inttrcst on hotli sidos. on 
the sums fallinpj due prior to the avera.pe date, and on those falling dtic subse- 
quently, and not by the interest on tliose falling due prior, and by tli« f 
on those fallinij due subsequent, as would be strictly correct. The ; 
against the creditor or holder of the demands, in like manner and n . . \ 
tent, as shown in note under Discount. 



EQUATION OF PAYMENTS. 133 

The following exhibits the face of an account in the ledger, and 
frhe time (date) at which it averages due is required. 



360 


, April 10 $250.26 — 6 mo. 


Due Oct. 


10. 




June 25 320.56 — 6 '' 


'' Dec. 


25. 




July 10 50.62 — 3 *^ 


" Oct. 


10. 




Aug. 1 210.84 — 4 '' 


'' Dec. 


1. 




" 18 73.40 — 5 " 


'' Jan. 


18. 




Oct. 15 100. — cash 


'' Oct. 


15. 



Example. — Practical method of stating and working. 
1860. Due Oct. 10, $301 
" " Dec.25, 321 X "^5 =■ 24396. 
** ** " 1, 211 X 52= 10972. 
'' " Jan. 18, 73 X 100 = 7300. 
** ** Oct. 15, 100 X 5 = 500. 

TOOO" ) 43168(43 days, = Nov. 22, 1860. 

Arts, 

COMPOUND AVERAGE. 

Compound Average consists in finding the time at which the hal- 
ance of an account or demand averages due, whose sides — the debit 
and the credit — average due at different dates. 

Rule. — Multiply the less sum or side by the difference in days 
between the two dates — that at which the debit side averages 'due 
and that at which the credit side averages due — and divide the prod- 
uct by the difference of the sums or sides ; the quotient will be the 
number of days that one of the dates must be set back, or the other 
forward, to mark the time sought ; for which last, 

special rule. 
Earlier date with larger sum, set back from earlier. 
Later date with larger sum, set forward from later. 

Example. — The debit side of an account in the ledger foots up 
$400, and averages due Oct. 12, 1860 ; the credit side of the same 
account foots $300, and averages due Nov. 16, 1860. At what date 
does the balance or difference between the two sides average due ? 

400 300 

300 35 

100 ) 10500 ( 105 days earlier than Oct. 12, = June 29, 1860. Ans, 

Example. — The debit side of an obligation foots $250, and aver- 
ages due May 17, 1860 ; the credit side of the same obligation foots 
$175, and averages due May 1, 1860. At what date does the differ- 
ence of the sides average due 1 
250 175 

175 16 

75 ) 2800 ( 37i days later than May 17, = June 23, 1860. A?is. 
12 



134 GENERAL AVERAGE. 



GENERAL AVERAGE. 

It is the established usage that whatever of either of the three 
commercial interests — the ship, the cargo, or the freight — is 
voluntarily sacrificed or destroyed for the general good, or with 
the view of saving the most that may be saved when all is in immi- 
nent danger of being lost, is matter of general loss to the respec- 
tive interests, and not more especially to the interest voluntarily 
abandoned than to the others. »So, too, the losses and damages inci- 
dent to the voluntary sacrifice, and collateral therewith, together 
with the expenditures which the master has been compelled to 
make for the general good, in consequence of disaster, are matters 
of general average, or are to be contributed for, pro rata^ by the 
several interests. 

The contributory interests are the ship, the cargo, and the 
freight, at their net values, independent of charges, premiums 
paid lor insurance, &c. 

The contrii)ut(jry value of the ship, generally, is her value at the 
port of departure at the time of leaving, less the premium paid for 
tier insurance. 

The contributory value of the cargo is its net .value, in a sound 
state, at the port of destination, if the voyage be completed; or its 
invoice vahie if the voyage be broken up and the cargo returned 
to the port whence it was shipped ; or its market-value at any in- 
termediate port, where of necessity it is discharged and disposed of. 
The value of the goods jettisoned, and to be contributed for, is 
their value after the same manner ; and that value is a part of the 
contributory value of the cargo, as well as a matter of general 
average. 

The contributory value of the freight, generally, is the gross 
amount or amount per freight-list, less onc-thinl part thereof, in 
most of the States ; but, in the State of New York, one-half thereof, 
for seamen's wages and other expenses. The loss of freight by 
jettison, when any freight is earne«l, is matter of general average. 
If the cargo is transshipped on board another vessel, and in that 
way sent to the port of destination, the contributory value of the 
freight is the gross amount, less the sum paid the other vessel. 

The voluntary damage to the ship, with a view to the general 
good, — such as throwing over her furniture, destroying her equip- 
ments, cutting away her masts, breaking up her decks to get at the 
cargo for the purpose of throwing it over, &c., — is contributed for 
at two-thirds the cost of repairing and restoring ; the new articles 
being supposed one-half better, or worth one-half more, than the 
old. 



GENERAL AVERAGE. 135 

If we let V = contributory value of tlie vessel, 
C 1= contributory value of the cargo, 
F z:r contributory value of the freight, 
d zz: aggregate amount of losses to be averaged, then 
cZ -^ (V -f- C -f- F) zr r, the per cent, of each interest that each 
must contribute, and 

Vx ?' = VesseFs share of the loss, 
C X ^ ^= Cargo's share of the loss, 
Fx r=: Freight's share of the loss. 
When a contributory interest's share of the loss is to be distrib- 
uted among the several owners of that interest, the same pro rata 
method is to be observed : thus 

A X ^ = sum A must contribute, 
B X ^= sum B must contribute, 
D X ^ = sum D must contribute ; 
A, B, and D being A's, B's, and D's respective shsures in that 
interest. 



136 ASSESSMENT OF TAXES. — INSURANCE. 



ASSESSIMENT OF TAXES. 

G = amount of taxable property, real and personal, as per 
grand list. 

A = amount of money to be raised, includlnp; the whole poll-tax. 

T = amount of money to be raised on property alone. 

n ::= number of ratable })olls. 

h = poll-tax per head. 

r =z rate per cent, to be raised on taxable property. 

P r=z an individual's taxable property, as per grand list. 

b =P'8 poll-tax. 

T = A — hi. r = T-^G. P r -f /; = Fs tax, including poll. 



INSUR.VNCE. 

Insurance is a written contract of indemnity, called the policy, 
by which one party (the insurer or underwriter) en^^ages, for a 
stipulated sum, called the premium (usually a per cent, on the 
value of the property insured), to insure another against a risk or 
loss to which he is exposed. 

Let P= Principal, or amount insured on, 
r =: rat4.' per cent, of insurance, 
a = premium for insurance. 

a = Pr. r = a -j- P. P = a -^ r. 

Example. — What is the premium for insuring on $4500 at 1^ 
per cent. ? 

4500X.^15 = S67.50. Ans, 



LIFE-INSURANCE. 

Life-insurance is predicated upon the oven chance in years, 
called the expectation of life, that an individual in general health 
at any given age appears by the rates of mortality to have of living 
beyond that age. 

The Carlisle Tables of Expectation, column C in the following 
tables, are used almost or quite exclusively in England, and by 
some insurance-companies in the United States ; while those by 
Dr. Wigglesworth, column W, computed with special reference to 
the rates of mortality in this country, are used by others. 

The Supreme Court of Massachusetts has adopted the Wiggles- 



LIFE INSURANCE. 



137 



worth rates of expectation in estimating the value of life-annuities 
and life-estates. 

TABLE 

Of Ages and 'Expectations from Birth to 103 Years. 



Age. 


c. 


w. 


Age. 


c. 


w. 


Age. 
52 


c. 


w. 


Age. 


c. 


w. 
6.59 





38.72 


28.15 


26 


37.14 


31.93 


19.68 


20.05 


78 


6.12 


1 


44.68 


36.78 


27 


36.41 


31.50; 


53 


18.9719.46} 


79 '5.80 


6.21 


2 


47.55 


38.74 


28 


35.69 


31.08 


54 


18.28il8.92 


80 5.51 


5.85 


3 


49.8-2 


40.01 


29 


35.00 


30.66 


db 


17.58|18.35 


81 5.21 


5.50 


4 


50.76 


40.73 


30 


34.34 


30.25 


56 


16.89 17.78 


82 1 4.93 


5.16 • 


5 


51.25 


40.88 


31 


33.68 


29.83 


57 


16.21 17.20 


83 j 4.65 


4.87 


6 


51.17 


40.69; 


32 


33.03 


29.43 


58 


15.55 16. 63i 


84:4.39 


4.66 


7 


50.80 


40.4 7i 


33 


32.36 


29.02 


59 


14.92a6.04 


85j4.12 


4.57 


8 


50.24 


40.14! 


34 


31.68 


28.62| 


60 


14.34 15.45 


86 3.90 


4.21 


9 


49.57 


39.72' 


35 


31.00 


28.22| 


61 


13.8214.86' 


87| 3.71 


3.90 


10 


48.82 


39.23: 


36 


30.32 


27.78 


62 


13.31 


14.26; 


88 3.59 


3.67 


11 


48.04 


38.64' 


37 


29.64 


27.34| 


63 


12.81 


13.66' 


89 3.47 


3.56 


12 


47.27 


38.02; 


38 


28.96 


26.91| 


64 


12.30 13.051 


90:3.28 


3.43 


13 


46.51 


37.41' 


39 


28.28 


26.47 


65 


11.79 12.43 


91 3.26 


3.32 


14 


45.75 


36.79 


40 


27.61 


26.04 


QQ 


11.27 11.96 


9213.37 


3.12 


15 


45.00 


36.17! 


41 


26.97 


25.61 


67 


10.7511.48' 


93 1 3.48 


2.40 


16 


44.27 


35.76; 


42 


26.34 


25.19j 


68 


10.23 


11.01! 


94 1 3.53 


1.98 


17 


43.57 


35.371 


43 


25.71 


24.771 


69 


9.70 


10.50 


95 1 3.53 


1.62 


18 


42.87 


34.98 


44 


25.09 


24.35' 


70 


9.18 


10.06 


96 3.46 




19 


42.17 


34.59 


45 


24.46 


23.92: 


71 


8.65 


9.60, 


9713.28 




20 


41.46 


34.22 


46 


23.82 


23.37 


72 


8.16 


9.14( 


98 1 3.07 




21 


40.75 


33.841 


47 


23.17 


22.83 


73 


7.72 


8.691 


9912.77 




22 


40.04 


33.461 


48 


22.50 


22.27 


74 


7.33 


8.25 


100 


2.28 




23 


39.31 


33.08 


49 


21.81 


21.72 j 75 


7.01 


7.83| 


101 


1.79 




24 


38.59 


32.701 


50 


21.11 


21.17 


1 76 


6.69 


7.40l 


102 


1.30 




25 


37.86 


32.33| 


51 


20.39 


20.61 


77 


6.40 


6.99 


103 


0.83 





Thus, by the tables, a man in general good health at 21 years of 
age has cgi even chance, by the Carlisle rate of mortality, of living 
40| years longer; by the Wigglesworth rate, of living 33^^^^^ 
years longer. So a man in general good health, at 60 years of 
age, has, by the Carlisle rate, an even chance of living 14.34 years 
longer; by the Wigglesworth rate, an even chance of living 15.45 
years longer, etc. 

12* 



138 



FELLOWSHIP. 



FELLOWSHIP. 



Fellowship calls for the dislribiilion of a given effect to each 
of the several causes associated in its production, proportional to their 
respective magnitudes one with another. 

It is a rule, therefore, adapted to the use of partners associated in 
business, in achicvino^ a/>ro rata distribution among themselves as indi- 
viduals, of the profits or losses pertaining to the company. 

Rule. — Multiply each partner's investment or share of the capita! 
stock, by the whole gain or loss, and divide the product by the sum 

of all the shares, or gross capital. 

Example. — Three men, A, B, and C, enter into partnership. A 
invests $500, B $700, and C S300. They trade and gain $400. 
What is each partner's share of the profits? 



A, $500 

B, 700 

C, .300 



500 X 100 -^ 1500 = $13.3. 3.3i = A's share. 
700 X 400 -^ 1500 = ISO.OOii = B's *' 
300 X 100 -^ 1500 = 80.00 = C's ** 

$1500 = gross capital. $400.00 Proof. 

Example. — D's investment of $000 has been employed eight 
months ; F/s, of $500, five months ; and F's, of $300, five months ; 
the profits of the company are $500, and arc to be divided pro rata 
among the partners. What is each ])artner's share? 

D, $600 X ^ = 4«^00 X i>00 -^ SSOO = $272.73, D's share. 

E, 500 X ^y = 2500 X 500 -^ 8S00 = 112. 05, E's ** 

F, 300 X 5 = 1500 X 500 -T- 8800 = 85.22 , F's " 

8800 $500. Proof. 

Example. — Of $120 distributed, there were given to A, J ; to B, 
J ; to C, -J ; and to D, ^, and there was nothing remaining. 
What sum did each receive ? 

\ of 120 = 40 X 120 ~- 111 = $42/j^ = A's share. 
\ of 120 = 30 X 120 -^ 114 = 31 jI = B's ** 
J of 120 == 24 X 120 -^ 114 == 25y^9 = C's *' 
I of 120 = 20 X 120 -^ 114 = 21-jJg. = D'a " 
"TTI $120. Proof. 

Example. — Divide the number 180 into 3 parts, which shall 
be to each other as 2, 3, 4. 

4 of 180 = 90 X 180 -^ 195 = 83.08 
I of 180 r= 60 X 180 -f- 195 = 55.38 
\ of 180 = 45 X 180 -f- 195 = 41.54 

195 180.00 Proof. 



ALLIGATION. 139 

Example. — $400 are to be divided between A, B, and C, in 

tbe ratio of j- to A, J to B, and ^ to C ; how much will each 
receive ? 

4 of 400 = 200, and 200 X 400 -i- 500 =8160 =: A's share. 

I of 400 = 200, and 200 X 400 -j- 500 = 160 r= B's share. 

I of 400 = 100, and 100 X 400 -^ 500 = 80 = C's share. 

500 • S400. Proof. 



ALLIGATION. 

Alligation Medial is a method by which to find the mean price of 
a mixture or compound, consisting of two or more articles or ingre- 
dients, the quantity and price of each being given. 

Rule. ; — Multiply each quantity by its price, and divide the sum of 
the products by the sum of the quantities ; the quotient will be the 
price per unity of measure of the mixture ; and, having found the 
price of the given quantities as mixed, any quantities of the same 
materials, taken in like proportions, will be at the same price. 

Example. — If 20 lbs. of sugar at 8 cents, 40 lbs. at 7 cents, and 
80 lbs. at 5 cents per pound, be mixed together, w^hat will be the mean 
price, or price per pound, of the mixture? 

20 X 8 = 160 

40 X 7 = 280 

80 X 5 = 400 

140 ) 840 ( 6 cents. Ans, 

The several kinds, then, at their respective prices, taken in the 
proportion of 1 at 8, 2 at 7, and 4 at 5 cts., w^ill form a mixture worth 
6 cts. a pound. 

Example. — If 10 lbs. of nickel are w^orth $2, and 24 lbs. of copper 
are worth $44, and 8 lbs. of zinc are worth 40 cts., and 1 lb. of lead is 
worth 5 cts., what are 5 lbs. of pretty good German silver worth? 
(2.^+A5i)^+4o 4_5IKA = 81 cents. Ans. 

Alligation Alternate is a method by which to find what quantity 
of each of two or more articles or ingredients, whose prices or quali- 
ties are given, must be taken to form a mixture or compound that 
shall be at a given price or of a given quality between the two 
extremes. It also applies to the finding of relative quantities when 
the quantity of one or more of the articles is limited. 

Kule. — Connect the given prices or qualities — a less than the 
given mean with that one or either one that is greater — and to the 
^tent that all be thus connected ; then place the difference between 



140 



ALLIGATION. 



each given and the given mean opposite, not the given, or tho 

fiven mean, but the given with which it is al heated ; the num* 
er standing opposite each price or quality will 1x5 the quantity that 
must be taken at that price, or of that quality, to form a mixture or 
compound at the price or of the quality desired. And, being propor- 
tions respectively to each other, they may be taken in ratio greater 
or less, as desired. 

Example. — In what proportions shall I mix teaa at 48 cents a 
pound and 54 cents a pound, that the mean price may be 50 cents a 

pound ? 

In Ihc pmportloiia 

.^5 48-1 5 4 lbs at 48 cts. > 
^" ) 54J ^ 2 lbs. at 54 cts. J 
Or, as 2 at 48 to 1 at 54. 



Ans. 



preof. H 5 ;»+'=< ='-™- 



'l 



3 X 50 



150. 



ExAMPLK. — In what ]>n)p()rtions shall I mix leas at 48, 54, and 79 
cents a pound, that the nuxture may average CO cents a pound ? 

( 48"^ rj. 



60 { 54 



V2 at 48 ) (2 at 48 ) 
12 at 54 }= < 2 at 5.1 S 
18 at 72 ) (3 at 72 ) 



Ans. 



( 72il 12 + 6, 

Example. — A wine dealer has received an order for a quantity of 
wine at 50 cts. a gallon. lie has none rrady mannfacturrd at that 
price. lie has it at 40 cts., at 50 cts., and at 80 cents a gallon, and 
he has water that cost him nothing. He wishes to fdl the order 
with a mixture composed of the four materials — the water and the 
three dilferont priced wines. In what proportions must he mix them, 
that the mean or average price may be 50 cents ? 



Or, 50 



= 90 gals. 
Ans. 

fOOnl Vy + 30" 
40-11 30 
56-^1 50 
80J 50 + 10 




Or, 50 



Or, 50 





Ans. 


30 


= 30"! 


^> + 


30 = 30 


10 


= 10 ' 


50 + 


10 = 60^ 




= 13G gals. 




Ans, 


00- 


1 ^ "1 


)J'^^i 


\ -f 30 1 
1 50 -f 10 ' 


n 5rJ 


^80- 


J 10 



= 170 gals. 



= 112 gals. 



If, now, having found the proportions desired, it is wished to limit 
one of the articles in quantity — say the best wine to 8 gallons in the 



i 



rNVOLTJTION — EVOLTTTIOW. 141 

mixture — the pioportions of the remaining articles thereto are found 
thus : — 

Instance, 1st example, — 

iA • Q •• QH OA I And the mixture will consist of 
IS ;?;;1"4\!8 + 40-f24 + 4| = 76| gallon,. 

If, instead, it is desired to mix a given quantity, say 100 gallons, 
and proportioned, say as in first example, the quantity to be taken of 
each is ascertained by the following 

Rule. — As the sum of the relative quantities is to the quantity 
required, so is each relative quantity to the quantity required of it 
respectively. 

The sum of the relative quantities alluded to is 6 + 30 + ^^ + 10 
■»96; then, 

96 : 100 :: 6 = 61 
96 : 100 :: 30=3ii 
96 : 100 :: 50 = 52jl 
96 : 100 :: 10=10^53 



INVOLUTION. 

Involution consists in involving, that is, in multiplying a number 
one or more times into itself. The number so involved is called the 
root, and the product arising from such involution, its power. 

The second power, or square, of the root, is obtained by multiplying 
the root 07ice into itself, as 4 X 4 = 16 ; 4 being the root and 16 its 
square. 

The third power, or cube, of a number, is obtained by multiplying 
the number twice into itself, as 4 X 4 X 4 = 64 ; and so on for any 
power whatever. 

When a number is to be involved into itself, a small figure called 
the indea; or exponent is placed at its right, indicating the number of 
times it is tabe so^ involved, or the power to which it is to be raised. 
Thus, 3^ = 3 X 3 X 3 X 3 = 81; and 4^ = 4 X 4 X 4 = 64. 



EVOLUTION. 

Evolution is the opposite of Involution. It consists in finding a 
root of a given number, instead of a power of a given root. 

When the root of a number is required or indicated, the number is 
WTTitten with the V before it : and the character or denomination of 
the root, if it be other than the square root, is defined by an index 



142 EvoLtmoN. 

figure placed over the sign. When the square root of a number i« 
required, the sign {a/) is placed before the number, but the index 
(2) is usually omitted. Thus, a/25, shows that the square root of 
25 is required, or to be taken ; and \/25 shows that the cube root 
is required. The operation is usually called extracting the root. 

TO EXTRACT THK SQUARE ROOT. 

Rule — 1. Separate the given number into periods of two figures 
each, by placing a point over the first figure, third ^ fij^^^ &c., counting 
from right to left — the root will consist of as many figures as there 
are periods. 

2. Find the greatest square in the left hand period, and place its 
root in the quotient ; subtract the s<piare of the root from the left 
hand period, and to the remainder bring down the next period for a 
dividend. 

3. Multiply the root so far found — the figure in the quotient — by 
2, for a divisor; see how many times the divisor is contained in the 
divided, except the right hand figure, and place the result (the num- 
ber of times it is contained) in the quotient, to the right of the figure 
already there, and also to the right of the divisor ; multiply the divi- 
sor, thus increased, by the last figure in the quotient, and subtract the 
product from the dividend, and to the remainder bring down the next 
period for a dividend. 

4. Multiply the quotient — the root so far found (now consisting of 
two figures) — by 2, as before, and take the product for a divisor; 
see how many times the divisor is contained in the dividend, except 
the right hand figure, and ])lace the result in the quotient, and to the 
right of the divisor, as betbre ; multiply the divisor, as it now stands, 
by the firjure last placed in the quotient, and subtract the product from 
the dividend, and to the remainder bring down the next i)eriod for a 
dividend, as before. 

5. Multiply the quotient (now consisting of 3 figures) by 2, as 
before, and take the product for a divisor, and in all respects proceed 
as when seeking for the last two figures in the quotient. The quo- 
tient, when all the periods have been brought down and divided, will 
be the root sought. 

NoTK. — 1. If ihere \b a remainder after findini? ihc Inlegcr of a root, annex period* of 
ciphers thereto, and proceed as when seeking for ihe integer. The quotient figures will 
be the decimal |xiriion of the nxit. 

2. If the given number is a decimal, or consists of a whole number and decimal, point 
off the decimal from lefl to rieht. hy placing the point over the second, fourth, sixth, *c, 
figures therein, and fill the last perio^l. if incomplete, by annexing a cipher. 

3. If the dividend does not contain the divisor, a cipher must be placeil in the quolienl. 
and also at the riuhl of the divisor, and the next period brought down ; then the diridena 
raust be divided by the divisor as increaseti. 

4. If the quotient fig\ire. obtained by dividing by the double of the root, is loo large, m 
will sometimes be thccase, (see 3d Example) it must be dropped, and a len — odo which 
is the true measure — taken in its stead. 



EVOLUTION. 143 



Example. — Required the square root of 123456.432. 
123456.4320 ( 351.3636+. Ans, 



65) 334 
325 



70 . ) 956 
701 



7023 ) 25543 
21069 



70266 ) 447420 
421596 



702723 ) 2582400 
2108169 



7027266 ) 47423100 
42163596 

5259504 

Example. — Required the square root of 10621. Also, of 28561, 



i062i ( 103.05+. Ans. 

1 



203 ) 00621 
609 



20605 ) 120000 
103025 



16975 



28561 ( 169. Ans. 
1 



26 ) 185 
156 



329 ) 2961 
2961 



TO EXTRACT THE CUBE ROOT. 

Rule — 1. Separate the given number into periods of three figures 
each, by placing a point over the first, fourth, seventh, &c., counting 
from right to left — the root will consist of as many figures as there 
are periods. 

2. Find the greatest cube in the left hand period, and place its root 
in the quotient ; subtract the cube of the root from the left hand pe- 
riod, and to the remainder bring down the next period for a dividend. 

3. Multiply the square of the quotient by 300, for a divisor; see 
how many times the divisor is contained in the dividend, and place 
the result (except that the remainder is large, diminished by one or 
two units) in the quotient. 

4. Multiply the divisor by the figure last placed in the quotient, 
and t(^he product add the square of the same figure, multiplied by the 
other figure, or figures, in the quotient, and by 30 ; and add also thereto 



144 



EVOLUTION. 



the cube of the same fi^re, and take the sum for the subtrahend ; stih- 
tract the subtrahend from the dividend, and to the remainder bring 
down the next period for a dividend, with which proceed as with the 
preceding, so continuing until the whole is completed. 

Note — 1. Decimals must be pninled from left to right, by placing a point over Um 
third, sixth, <fcc., fibres in that direction. 

2. If the tlivisnr is not contained hy the dividend, place a cipher in the quotient, and 
annex two ciphers to the divis<»r, and brin? down the next periotl for a dividend, and naa 
the divi5?or, a-s thus iiicrease<l. for finding the next quotient figure. 

3. If there is a remainder after finding the inleirer of the root, annex a peiiod of thrae 
ciphers thereto, ami priK:eed for the decimal of the root as if scckinf for Iho inlefer, an- 
nexiii? a pcri(Nl of thn?c cipticrs to each remainder until the decinud 18 carried to aa man/ 
places of figures as desired. 

Example. — Required the cube root of 47421875.0324. 

47421875.632400 ( 361.959-f . 
27 Arts. 

3^X 300 = 2700)20421 
6 

10200 

6"^ X 3 X 30 = 3'JIO 

G' = 210 =19656 

36^ X 300 = 388800 ) 765875 
1^ 

38HS00 
1^ X 30 X 30 = 10H(1 

1^ = 1=380881 

361^X300 = 39096300) 375994032 
9 

351866700 
9'^ X 301 X 30 = 877230 

9^= 729 = 352744659 



3619^ X 300 = 3929148300 ) 23249973400 
5 

19645741500 
52 X 3619 X 30 = 2714250 

5^= 125= 19648455875 

36195' X 300 = 393023107500 ) 3601517525000 

9 



3537210607500 
' X 36195 X 30 = 87953850 
93_^ 729 = 3537298622079 

64218902921 



EVOLUTION. 145 

Example. - - Kequired the cube root of 32768. Also, of 8489664. 



32768(32. 
27 Ans. 

32x300 = 2700 ) 5768 
2 

5400 
22X3X30=360 

23= 8 = 5768 



8489664(204. 
_8 Ans. 

22 X 300 = 120000 ) 489664 
4 

480000 
42 X 20 X 30 = 9600 

43 = 64=489664 



General Rule for extracting the roots of all powers, or for finding 
any proposed root of a given number. 

1. Point off the given number into periods of as many figures 
each, counting from right to left, as correspond Avith the denomina- 
tion of the root required ; that is, if the cube root be required, into 
periods of three figures, if the fourth root, into periods 0^ four fig- 
ures, &c. 

2. Find the first figure of the root by inspection or trial, and place 
it at the right of the number, in the form of a quotient ; raise this 
quotient figure to a power corresponding with the denomination of 
the root sought, and subtract that power from the left hand period, 
and to the remainder bring down the first figure of the next period, 
for a dividend. 

3. Raise the root thus far found (the quotient figure) to a power 
next inferior in denomination to that of the root required, multiply 
this power by the number or index figure of the root required, and 
take the product for a divisor ; find the number of times the divisor 
is contained in the dividend, and place the result (except that the 
remainder is large, diminished by one or two units) in the quotient, 
for the second figure of the root. 

4. Raise the root thus far found (now consisting of two figures) to 
a power corresponding in denomination with the root required, and 
subtract that power from the two left hand periods, and to the re- 
mainder bring down the first figure of the third period, for a divi- 
dend ; find a new divisor, as before, and so proceed until the whole 
root is extracted. 

Example.— Required the fifth root of 45435424. 

45435424(34. Ans. 
3^ = 243. 

. 34 X 5 ) 2113 
345=45435424 

13 



146 ABITHMETICAL PE0GBES8I0N. 

Example. — Required the fifth root of 432040. 0a54. 

432040.03540(13.4+. Ans, 

15 = 1 

P X 5 ) 33 
13-^ =« 371203 



13^X5) G07470 
13.4^ = » 43204003424 

116 

For instructions touchin;^ special cases, see Notks relative to tho 
extraction of the srjuare root, and to the extraction of tho cube root. 

The /s/ of the »/ of any numlxjr = /^ of that number 
** V of tho /^ = a/. 
" V of the V of the V = /C/- 
** ^ ofthe>s/ = >v/- 
** V of the Ai/ = ^, &o. 



AIUTIIMETICAL PROGKESSION. 

A series of three or more numlwrs, increasing or decreasing by 
equal difTorenccs, is called an arithmrtical profession. If the num- 
bers progressively increase, the series is called an ascendinir arith- 
rmfira/ proi^ression; and if they progressively decrease, the scries ifl 
calh^l a dcscendins^ arithmetical pro f^cssion. 

The numbers forming the series are called the terms of the 
pro^^ression, of which the first and the last are called the extremes^ 
and the others the means. 

The difiereiice between the consecutive terms, or that quantity by 
which the numbers respectively increase upon each ether, or dccreaao 
from each other, is called tlie commim difftrcncr. 

Thus, 3, 5, 7, 9, 11 , 1'tc, is an ascendinn; arithmetical progression, 
and 11, 9, 7, 5, 3, is a descending arithmetical progression. In these 
progressions, in both instances, 11 and 3 are the extremes^ of which 
11 is the s^rcatcr iwtrcmc, and 3 is the iess extreme. The numbers 
between these, (9, 7, 5,) are the means,. 

In every arithmetical progression, the sura of the extremes is 
equal to the sum of any two means that are equally distant from the 
extremes ; and is, therefore, equal to twice the middle term, when 
the series consists of an odd number of terms. Thus, in the fore- 
going series, 3 + 11=5 + 9 = 7X2. 

Tho greater extreme^ the less extreme, the number of terms ^ the 



ABITHMETICAL PROGRESSION, 147 

tommon difference, and the sum of the terms, are called the jfive prop- 
erties of an arithmetical progression, of which, any if Aree being given, 
the other two may be found. 

Let 5 represent the sum of the terms. 
'* E *' the greater extreme. 
" e ** the less extreme. 
*' ^ ** ' the common difference. 
" n ** the number of terms. 

The extremes of an arithmetical progression and the number of terms 
being given, to find the sum of the terras^ 

(E + e) X n 

2 = sum of the terms. 

Example, — What is the sum of all the even numbers from 2 to 
100, inclusive ? 

102 X 50 -r 2 = 2550. Ans, 

Example, — How many times does the hammer of a common dock 
strike in 12 hours 1 

(1 + 12) X 12 -r- 2 = 78 times. Ans. 
i — -7— + 1 ) X — i"""^ ^^^ ^^ *^® terms. 



{EX2 — n — IX^) X4^ = sum of the terms. 



{2e-\-n — IX^) Xi^ = sum of the terms. 

The greater extreme, the common difference, and the nvmber of terms 
of an arithmetical progression being given, to find the less extreme. 



E — (d X n — 1) = less extreme. 

Example. — A man travelled 18 days, and every day 3 miles far- 
ther than on the preceding ; on the hist day he travelled 56 miles ; 
how many miles did he travel the first day ? 



56 — (18 — 1 X 3) = 5 miles. Ans. 

— ( ) = less extreme. 

n V 2 / 

•^ X 2 — E = less extreme. 



148 ABITHMiniCAL PKOGBESSIOH* 

a/ (EX 2 + ^)'- — 5X ^X8 + rf = less extreme, when 
2 
V (2E-\-dy^ — 85^ is equal to, or greater than d. 



a/(2E-j-</)- — 85rfvy^</= less extreme, whm 
- 

V (2E-\-dy^ — Hsd is less than d. J 



>v/(2 e v>- ^/)- -|- 8 .w/ — ^ = greater extreme 


1 




dXn — I -\-c = greater extreme. 


1 


5 n — 1 X ^ 

^^-\- = greater extreme. 




2 5 -i- n — c = greater extreme. 





7^ c^ctrcmcs of an arithmetical yro^jcssion and the (tinuwm ntjcrcna 
being given J to find the nu/nher 0/ terms. 

E — r-f-^-|~l= number of terms. 

Example. — As a Iicavj Ixxly, falling freely through 8 pace, de- 
scends IG-j^ foot in the fir^t srcond uf its descent, 48-i^j feet in the 
next second, 80^^ in tlie third second, and so on; how m;uiy seo* 
onds had that Ixxly ]>eon falling, that descended 305/j feet in the 
last second of its descent ? 

305/g- — IG-rV = 289jJ -^ 32J = -f 1 = 10 seconds. Ans. 

W (2€^ d)''-{-S s d — d — c -^ d ~\-l=z numlxjr of terms. 



2 5-^E-f/v/(2 E-f-r/)t — 8 5r/ + ^= number of terms whea 

/y {2 E -\- dy- — H s d is equal to, or greater than d. 



2 5 -r- E + V' (2 E -j- r/)-' — % sd^d^s. number of terms wheo 
2 

V (2E-|-</)- — 8 5</ is less than d. 
5X2 



E+c' 



number of terms. 



ABITH^ETICAL PROGRESSION, 149 

The extremes of an arithmetical progression^ and the number of terms 
being given, to find the common difference. 

E — e 

7 = common difference. 

n — i 

Example. — One of the extremes of an arithmetical progression is 
28 and the other Is 100, and there are 19 terms in the series ; re- 
quired the common difference. 

100 ^A^ 28 -^ 1^ 19 = 4. Ans, 

fsX^ \ 

E — ^ -^- 1 ;^ I g — ^ / ^ common difference. 

2s-^w — 2e 

"t = common difference. 

n — i 

2E— (25-^n) 

-^ = common difference. 

n — i 

Example. — The less extreme of an arithmetical progression is 28, 
the sum of the terms 1216, and the number of terms 19 ; required 
the 7th term in the series, descending. 

1216 X 2 -^ 19 = 128 = sum of the extremes. 
128 — 28 = 100 = greater extreme. 
100 — 28 = 72 = difference of extremes. 



72 ~- n — 1 (18) = 4 = common difference. 
100 — (T^^^^l X 4) = 76 = 7th term descending. Ans. 
Required the 5th term from the less extreme, in an arithmetical 
progression, whose greatest extreme is 100, common difference 4, 
and number of terms 19. 



100 — (19 — 5 X 4) = 44. Ans, 

To find any assigned number of arithmetical means ^ between two given 
numbers or extremes. 

Rule. — Subtract the less extreme from the greater, divide the 
remainder by 1 more than the number of means required, and the 
quotient will be the common difference between the extremes ; 
which, added to the less extreme, gives the least mean, a'hd, added 
to that, gives the next greater, and so on. 



Or, E — e -7- 711 -}-- 1 = G^, E being the greater extreme, e the less 
extreme, m the number of means required, and d the common differ- 
ence. 

And^T^' f^-f 2f/, 6'+3r/, &c. ; or, E — tZ, E — 2^?, B — 3 ^, 
&c., Avill give the means required. 

13* 



150 OBOMETRICAL PR0GRI33I0N. 

ExAMFLB. — Required to find 5 arithmetical means between tbe 
numbers 18 and 3. 

18 — 3 = 15-T-G = 2A, and 
34-2i = 5i + 2i =8-|-2i = lOi 4-2.i = 13 + 2i = 15i. 
5i, 8, lOi, 13, 15^, therefore, are 5 arithmetical means, between the 

extremes, 3 and 18. 

Note. — The arithmetical mean between any two numbers nmy be found by dividing 
Ike sum of tbote numbers by2', thus, the arithmetical mean of 9 and Si9C94~^)~r^=-^* 






GEOMETRICAL TROGRESSION. 

A scries of three or more numbers, increasing by a common mul- 
tiplier, or decrciising by a C(jmmon divisor, is called a gcomriric/if 
frogrcssion. If the greater numbers of the prt>gre88ion are to tlio 
riglit, the progression is called an ascending gronuirical progression y 
but, on the contrary, if they are to the left, it is called a dcscrnding 
geometrical progression. The number by which the progression 19 
formed, that is, the common multiplier, or divisor, is called the 
rcUio. 

The numTxTS fjrming the series are called tlio tcrrns of the pro- 
gression, of ^vhicll the tirst and the last are callod the cTircrnes, and 
the others the means. The greater of the extremes is called tho 
greater extreme, and the less the less extreme. 

Thus, 3, 0, 12, 24, 48, is an ascending geometrical progression, 
because 48 is as many times greater than 24, as 24 is greater 
than 12, &c. ; and 250, 50, 10, 2, is a descending geometrical pro- 
gression, because 2 is as many times less than 10, as 10 i» less than 
• 50, &c. 

In the first mentioned series, (3, G, 12, 24, 48,) 48 is tho greater 
extreme, and 3 is the less extreme; the numbers C, 12, 24 are tho 
means in that progression. 

So, too, of the progression 250, 50, 10, 2 ; 2^>0 and 2 arc the ex- 
tremes, and 50 and 10 are the means. 

In the Ih-st mentiuncil progression, 2 is the ratio, and in the last, 
or in the progression 2, 10, 50, 250, 5 is the ratio. 

In a geometrical progression, the product of tho two extremes is 
equal to the product of any two means that are equally distant from 
the extremes, and, also, equal to the square of the middle term, 
when the progression consists of an odd num1)cr of terms. 

Thus, in the progression 2, 6, 18, 54, 162 ; 102 X 2 = 54 X ^ 
= 18 X 18. 

When a geometrical progression has but 3 terms, either gf the 



(GEOMETRICAL PROGRESSION, 151 

extr(5mer3 is called a third proportional to the other two ; and the 
middle term, consequently, is a mean proportional between them. 

Thus, in the progression 48, 12, 3, 3 is a third propo^^tional to 48 
and 12, because 48 divided by the ratio = 12, and 12 divided by the 
ratio = 3 ; or 3 X ratio = 12, and 12 X ratio = 48 : 12 is the mean 
proportional, because 12 X 12 = 48 X 3, 

Of the 5 properties of a geometrical progression, viz,, the greater 
esitreme, the less extreme, the number of terms, the ratio, and the sum 
of the terms, any three bein^ given, the other two may be found. 

Let s represent the sum of the terms, 

*' E ** the greater extreme. 

** e ** the less extreme, 

^' r ** the ratio. 

*' n '•* the number of terms. 

'' n when affixed as an index or exponent, represent that the 
term, number, or quantity, to which it is affixed, is to be raised 
to a power equal to the number of terms in the respecdve progres- 
sion, &c. 

Any tlvree of the five parts of a geometrical progression being given^ l# 
jfind -the ranaining two parts ^ 

E — e 

■ E = sum of the terms. 



r— 1 
EXr- 



r— 1 
r» X c — e 



■ sum of the terms. 



r— 1 
E— (E-T- r'^) 



= sum of tlie terms. 



^ 2 j-E = sum of the terms. 

E — e 



^^(E^e)-l 



-j- E = sum of the terms. 



{ 1 Example. — The greater extreme of a geometrical progression is 

1162, the less extreme is 2, and there are 5 terms in the progression ; 
required the sum of the series. 

162 — 2 

: 80 + 162 = 242, Ans, 



^(162~2)-1' 



sXr — l + e 

= greater extreme. 



152 OEOHETRICAL PROOKESSIOW, 

— X ^ = greater extreme. 
y.n-1 N^ c = greater extreme. 



r" — 1 



• greater extreme* 



s — (5 — E) X ^ = less extreme. 
E -r- r"""^ = less extreme. 



^Xr°-^Xr— 1 
3 — c 



=Ie88 extreme*, 



J — E 



= ratio. 



iXr-1 



•■;i/-« ratio. 



|- 1 = r^ ; 71, therefore, is equal to the ncHnbcr of timet 



tXr— 1 
that r mnst bo multiplied into itself to equal -\- 1. 



1 *^ 



.1— {5 — K)Xr 

Example. — A fanner proposed to a drover that ho would sell. 
liim 12 sheep and allo\y him to select them fn)m his flock, providedl 
the drover would pay 1 cent for the first selected, 3 cents for the 
second, 9 cents for the third, and so on ; what sum of money wo 
12 sheep amount to, at that rate? 

— T — = 5, then 

31- X 1 ~ 1 

3^3j — = S2G57.20. Ans, 

Note. — Ratio* , cubed = ratio^^ ; ratio®, squared = ratio'^, fcc. 

When it is required to fmd a high power of a ratio, it is conTcn-f 
lent to proceed as follows, viz. : write down a few of the lower or' 
leading powers of the ratio, successively as they arise, in a line, ono 
after another, and place their respective indices over them ; then 



^GEOMETRICAL PROGRESSION, 153 

will the product of such of those powers as stand under such indices 
whose sum is equal to the index of the required, power, equal the 
power required. 

Example, — Required the 11th power of 3, 

12 3 4 5 

3 9 27 81 243 

Here 5-{-44-2 = ll, consequently, 

243 X 81 X 9 = 11th power of 3, or 

5 X 2 -[- 1 = llj consequently, 
243 X 243 X 3 = 11th power of 3, ot 

4 X 2 + 3 = 11, consequently, 
81 X 81 X 27 = 11th power of 3, or 

3 X 3-[-2 = ll, consequently, 

273 X 9 = r^i = 177147. Ans. 

To find any assigned number of geometrical means, between two given 
numbers or extremes. 

Rule. — Divide the greater given number by the less, and from 
the quotient extract that root whose index is 1 more than the 
number of means required ; that is, if 1 mean be required, extract 
the square root ; if two, the cube root, &c., and the root will be the 
common ratio of all the terms ; which, multiplied by the less given 
extreme, will give the least mean ; and that, multiplied by the said 
root, will give the next greater mean, and so on, for all the means 
required. Or the greater extreme may be divided by the common 
ratio, for the greatest mean ; that by the same ratio, for the next 
less, and so on. 

Example. — Required to find 5 geometrical means between the 
numbers 3 and 2187. 

2187 -T- 3 = 729, and /^n^ = 3, then — 
3X3 = 9X3 = 27X3=81X3 = 243X3 = 729, that is, the 
numbers 9, 27, 81, 243, 729 are the 5 geometrical means between 
3 and 2187. 

Note. — The geometrical mean between any two given nmnbers is equal to the square 
root of the product of those numbers. Thus the geometrical mean between 6 and 20, =a 
V(5X20) = 10. 



154 ANNUITIES, 



ANNUITIES. 

An annuity, strictly speakinpr and practically, is a certain join 
of money by the year ; payable, usually, cither in a single pay- 
ment yearly, or in half, lyilf-yearly, quarter, quarter-yearly, &c., 
and for a succession of years, greater or less, or forever. Pensions^ 
awardsy ber^uests, and the like, that arc made i>ayablc in fixed 
sums for a succession of ]>aymenls, are commonly rated by the 
year, and denominated annuities, 

A cuiTcnt annuity that has already commenced, or that is to 
commence after an interval of time not greater than that between 
the stipulated payments, is said to be m jx^ssrssion. 

One that is to commence or cease on the occurrence of an 
indeterminate event, as ujK)n the death of an individual, is a rc- 
versionary^ ccntimjcnt^ or life aniuiity. 

One that is to commence at a given period, and tocontinne for a 
given number of years or ])ayments, is a certain annuity. 

One that is to continue from a given time, forever, is a perpetual 
annuifi/^ or a pcrpetuitij. 

Annuity paymentg do not exist fractionally : they mature, and 
exist only in that state, and are then due. 

A current annuity coiumenccs with a payment, and terminates 
with a payment. 

One current in the past is measured from a present included pay- 
ment, closes with an included payment, and is said to be in arrears 
or forborne y from a supix)sed cancelled payment one regular inter- 
val or time Ix'vond. 

One current in the future is measured from the present to the 
first included payment of the series, and from thence is said to con- 
iinuc to the close ; but if the interval from the present to the first 
included payment is ecjual to that between the successive pay- 
ments, it is supposed to continue from the present. 

Annuities in negotiation are adjusted, with regard to time, by 
interest y or (liscount, or both. 

The TABLES applicable to comi>ound interest and compound dis- 
count are applicable In adjusting annuities at compound rates. 

To find the Avwunt of a Current Annuity in Arrears. 

Lemma. — The amount of an annuity that has been forborne 
for a given time is equal to the sum of the several payments that 
have become due in that time, plus the interest on each, from the 
time it became due, until the close of the time. 



ANNUITIES. 155 

Then the amount of an annuity of SlOO, payable in a single 
payment annually, but delayed of payment 4 years, allowing sim- 
ple interest at 6 per cent on the pa} ments, is 

100 X 1.18— 118 
100 X 1.12— 112 
100 X 1.06= 106 
100 X 1 = 100 =i$4:SQ. 

And at 6 per cent, compound interest on the payments, it is 

100 X (1.06)3=: 119.10 
100 X (1.06)2=: 112.36 
100 X (1.06)1=106.00 
100 X 1 = 100.00 =S437.46. 

At 6 per cent, simple interest, when payable in half, half- 
yearly, it is 

50X1.21=60.50 
50 X 1.18 = 59.00 
50 X 1.15 = 57.50 
50 X 1.12 = 56.00 
50 X 1.09 = 54.50 
50 X 1.06 = 53.00 
50 X 1.03 = 51.50 
50 X 1 =50.00 = $442. 

And at 6 per cent, compound interest per annum, when payable 
in half-yearly instalments, it is 

50 X (1.06)8 X 1.03 = 61.34 
50 X (1.06)3 =59.55 

50 X (1.06)2 X 1.03z=:57.86 
50 X (1.06)2 =56.18 

50 X (1.06)1 X 1.03iiz54.59 
50 X 1.06 =53.00 

50 X 1.03 =51.50 

50 X 1. zz: 50.00 = $444.02. 

From the foregoing, we derive the following general Rules : — 

Let P :=. annuity or yearly sum, 

r = rate of interest per annum, 

a:=:rate of discount per annum, 

nov^'ziz nominal time of the annuity in full years, 

A = cmnount for the full years, 

D =z present wortJi for the full years. 



156 - ANNUITIES, 

When the annuity is payable in a sin^e payment jezriy^ 

A = Pn (l + ^^^V^)' ^™ple Interest. 

A= P — ~ t Compound Interest. 
When payable in equal half-yearly instalments, ■ I 

A = Fn{l-}.'-^^ + -^)y Simple Interest 

Air^PX ^ ^"^T~^ x(l + J)i Componnd Interest 

When payable in equal third-yearly instalments, 

A = Pn(l-f ^^^-f y)» Simple Interest- 
A = P — -^^ (i_|_ J)^ Compound Interest. 

When payable in quarter-yearly instalments, 

A = Prj ( 1 + ''-^^+ ^ ), Simile Interest 
A = P ^l±!2!lJ ( 1 -I- -^ ), Compound Interest. 

When there are odd payments, to find the amount, S. 

When 1 half-yearly, S = A(l 4- 4 r) 4- AP. 

1 third-yearly, S = A(l -- ^ r) -- ^P. 

2 '' ' S=A(l4-iir)-.4P(l + Jr) + JP 

= A(l —ir)-- r(<3 +r) -^ 0. 

1 quarter-yearly, S = A(l — I - - i^*- 

2 " S = A(l--;r)--P(8-f r)~ir,. 

3 " S=A(l4--]r)4-l'(-'^4-!0-^'*- 
For any number of equal and regular payments at compound 

interest per interval between the payments, S = P' ( :: — )y and 

for any nunilxr of equal and roi:^i}ar payments at simple interest 
per interval between the payment*;, S=P';i' (l -f- '"*"^~!J) ; ?' 

being a payment, n' or "' the number of payments, and r' the rate 
of interest per internal between the payments. But this. must not 
be confounded with compound interest annuaUr^, on ]>aymentA oc- 
curring semi-annually, quarterly, &c. 

Example. — AVhat is the amount of an annuity of SI 50, paya- 
ble in half, half-yearly, but delayed of payment 2 years and 72 
days, allowing compound interest jx^r annum at 7 per cent. ? 

150x^^—^^ = 8310.50, the amount foi- 2 years, if payable 
in yearly payments, and 



ANNUITIES. 157 

B10.5GX (l-^) =^$315.93, the amount for 2 years, if payable 
in half-yearly payments, and 

315.93 X 0-^^^) =: $320.29, the amount for 2 years and 72 
days, if payable in half-yearly payments. Ans, 

Example, — What is the amount of an allowance, pension, or 
award, of SlOO a year, payable quarterly, but forborne 3^ years, 
interest compound. per annum at 6 per cent. V 

100 X ^^'^^Jg'^ x(l+^^)==S325.52,the amount for 3 years, 

and 

325,52 (l4-.03) + 100x8.06-f-16=:S385.^6. Ans. 

Example. — What h the amount of $100 a year, payable in 
quarterly payments, and in arrears 4 years, interest being com- 
pound per quarter-year, at 6 per cent, a year ? 

25 [(1 + x/^— ^] X :^ • % tabular powers of (1 + r), page 

125, =$448.30. A71S, 

To f^nd the Present Worth ef an Annuity Current 

Lemma. — The present worth of an annuity that is to continue 
for a given time is equal to that sum of money, which, if put at 
interest from the present time to the close of the payments, will 
amount to the amount of the payments at that time ; and therefore, 
the times being full, is equal to the sum of the several payments, 
discounted, respectively, at the rate of interest for their respective 
times. 

Note. —If the foregoing preposition is tennble, it follows, since simple 
interest is due and payable annually, that the true present worth of an annuity 
having more than one year to run cannot be found by simple interest and dis- 
count. By simple interest and dis-count, at 6 i)er cent., predicating the rule 
upon tire foregoing lemma, the amount of $100, payable annually, and in 
arrears for 4 years, is $i363 and t\).Qpr£S^nt worthy at 6 per cent., is 

100 100 , 100 , 100 ^^^^ 

1- \ 4 = $349. 

1.24^1.18 1.12^1.06 ^ 

But $349 at 6 per cent, interest for 4 years, with the payments of int-erest annu- 
ally, will amount to $440.60; and at interest simply for 4 years it will amount 
to only $432..76. 

Then the present worth of an annuity of $100, payable in a 
single payment yearly, and to continue 4 years, or to become due 
1, 2, 3, and 4 years hence, interest and discount being compound 
per annum, and each at 6 per cent, zz: 
14 



158 AlfNUITI£0. 

P p p P 

(1+r)* + (T+7)» + (l+r)» + f+>— '^^-^^ = 

100 X (1.06)^1=119.10 

100 X (1.06^=112.36 

100 X (1*06) =106.00 

100 X 1 = 100.00 = 487.46 -f. (1.06)* zi: $346.51. 

And Interest at G per cent, and discount at 10, both compound, it b 
100 X (1.06/= 119.10 
100 X (1.06)»=: 112.36 
100 X 1.06 =106.00 
100 X 1 = 100.00 = 437.46 -^ (1.10)* = $298.79. 

Therefore, when the annuity is payable in a wngle payment 
yearly from the present time, 

D = r -T-»= ~ D whfii r and a are cnuaL 

When payable in half-yearly payments, 

D = Px^'^'*^'"JX(l + ir). 

r(l + a) 

When payable in third-yearly payments, 

D =:: rxC(l->-r)*~nx(l-Hr> ^ 

ni + a)' 
When payable in quarter-yearly payments, 
D s=: PC( 14T)'-I ) (l-Hr> , 
rtl + a)' 

When there arc odd paj-ments, to find the present worth, 8- 
There being a half-yearly, 8 = ,^^-^ + ,^. 
- 1 third-yearly, S = ,^>^+A^-. 

»( o u C _1> I 2r(14-ir) , 

^~*l-hla I 8(l-f|a) 

1 quartcr-yearly, ^ = f^j,+ r^Ta 
« 2 " s — ^ U- ^S^±^. 

For any number of equal payments, at equal intervals between 
the payipcnts, S = P' X -' ^'^'\~ r ; P' being a payment, °' the 



ANNUITIES. 159 

number of payments, and r' and a' the rates per interval between 
the payments. 

Note. — Since ^ ' — — - is the co-efflcient of P, for its present worth, at 
r(l + a) 
compound interest and discount, for the time » , at the rates r, a, it follows 
that tables of co-efficients of P for its present worth, at given rates, for any 
number of years, may be easily made. Thus (1.06* — 1) -M.06* x. 06 = 3.46511, 
the co-efficient of an annuity, P, for 4 years* continuance, interest and discount 
being compound per annum, at 6 per cent.; and (1.06^— D-j- (1.06' X-06) = 
1.83339, the co-efficient for 2 years, &c. 

If the annuity is deferred, then the difference of two of these co-efficients 
(one of them that for the time deferred, and the other that for the sum of the 
time deferred and the time of the annuity) will be the co-efficient of P for 
its present worth. Thus .3.46511 — 1.8.3.3.39 = 1.63172, the co-efficient of an annuity, 
P, for its present worth, when it is to commence two years hence, and to con- 
tinue 2 years, interest and discount being compound per annum, at 6 per 
cent, each; or D= 1.63172 P. 

In like manner, tables of other co-efficients, such as the formulae suggest, 
may be made that will greatly assist in calculating annuities. 

Example. — What is the present worth of an award of S500 a 
year, payable in half-yearly instalments, the 1st payment to mature 
6 months hence, and the annuity to continue three years ; interest 
and discount being 7 per cent., compounded yearly ? 

500X[(1.07y— 1]X (l.f) 

.07 X (1.07)3 ^ zz: $1335.13. Ans, 

Example. — What is the present worth of an annuity of $100, 
payable in half-yearly payments, and to continue 1^ years; interest 
and discount being 6 per cent, per annum ? 

100X[1.06-.l]Xl4^ 

D — 95 75.5 and 

-^— .06X1.06 — yo./oo, 

95.755 , 50 

1703- + ro3 = ^^^^-^^' ^^'- 

Example. — What is the present worth of an annuity of $500, 
payable in semi-annual instalments, and to continue 10^ years, 
interest and discount being compound per annum, the former at 6 
per cent., and the latter at 8 ? 

500C(1.06)"-l](l.f) 500 

.06(1.08r(l.f) +2(1.1) '■ 

A 250 - 

LOSiox 1-04+ LOT— ^'**- 



IGO ANNUITIES. ; 

By tabular powers of 1 -}- ^? P^c 125 : — 

500 y 79085 

- ' = S3052.G4, the present worth for 10 years* con- 

tinuancc, if payable in yearly payments, and 

3052.64 X 1.015 = S3008.43, 

the present worth for 10 years* continuance, if payable in half- 
yearly payments, and 

3098.43 -7- 1.04 -(- 500 -f- 2 X 1.04 =8321 9.04. An,<. 

When the interval of time from the present to the Ist paymeut 
is shnrtrr tlian that between the consecutive payments, and the 
annuity is payable in a single payment yearly, 

^ P[0+r)--l](l+^) , 

A=: ^ -. and 

r 

,^_ A P[0+r) --l3(l+.^-) 

d beiniT tlic time in days from the present to the 1st payment. 

So, if the annuity is payable in half-yearly, thinl-yearly, or quar- 
ter-yearly instalments, multiply by 1 -i- -J r, 1 -{- i '*» or 1 -f | r, M 
beibre directed; and if there are o<ld p.ivnunt^ t»nw.'i..1 for the 
present worth, S, as already directed. 

ExAMPLK. — Requirecl the present worth of an annuity of SlOO, 
payable yearly, to commenee 4 months hence, and to continue 4 
years ; interest and discount being G per cent annually, 

100 X (l.OG^ — 1) X (l-^o-) 

.06 X 1.06^ X (i.-^i!/— -^) 

To find the Present Worth of a Deferred Current Annuity ^ or of 
an Annuity in Reversion. 

^^^len the annuity is j)ayable in a single payment yearly, and the 
deferred time embraces full years only, 

D =r P -^ — -^— ^-7— TrrTT » n' beinj:: the deferred time. 

If it Is parable in half-yearly, third-yearly, or quarter-yearly 
instahnents, multiply by l'+ fr, 1 -|- i r, or 1 + f r, as already 



ANNUITIES. 161 

directed ; and, if there are odd paTments, find the present worth, S, 
as already directed. 

Example. — What is the present worth of an annuity of S150, 
payable yearly, to commence 2 years hence, and to continue 4 
years ; interest and discount being compound per annum, at 6 per 
cent. ? 

150 X (1.06* -— 1) -r .06 X 1.06« =z S462.59. Ans. 

Example. — Required the present worth of an annuity of $500, 
payable in semi-annual instalments, to commence 2^ years hence, 
and to continue 6 years ; allowing compound interest and discount 
annually at 7 per cent. 

500 X (1.07«— 1) X 1.— 

^^^ = 82046.44. Ans. 

.07 X 1.07« X 1.^ 

Example. — Required the present worth of an allowance, 
pension, or award of $125 a year, payable in half every half-year, 
to commence 7 months 24 days hence, and to continue 6|- years ; 
interest and discount being compound per annum at 5 per cent. 

125 X(1.05--1)X 1.0125 125 

.05 X 1.05^ X 1.03247 X 1.025^ 2 X 1.025 

^^ ^^^ 05 X^rToly"^"' — ^^^^-47, the present worth for 6 

years' continuance, if payable in yearly instalments ; and 

634.47 X 1.-^ = $642.40, 

the present worth for 6 years' continuance, if payable in half-yearly 
instalments ; and 

642.40 ^(l + :«5|^^) = 622.20, 

the present worth for 6 years' continuance, if payable in half-yearly 
instalments, and to commence 7 months, 24 days hence ; and 

622.20 4- (1 +-|) + ~^ = S668. Ar,s. 



Tojind the Present Worth of a Perpetuity, 

Lemma. — The present worth of an annuity to commence one 
year hence, and to continue forever, is expressed by that sum of 
^ money whose interest for 1 year is equal to the amount of the 
14* 



J|62 ANNUITIES. 

annuity for 1 year ; and so, pro rata, for perpetuities otherwise 
regularly alTected. 

Then when the annuity is to commence 1 year hence, and b 
payable in a single payment yearly . . . D = P -7- r. 

Payable in half-yearly instalments . . D =z ^ . 

Payable in third-yearly instalments . . D= . 

Payable in quarter-yearly instalments . i) = -- 

Exampm:. — What is the present worth of a perpetuity of SI 50 
a year, payable in a single payment yearly li-om the present time ; 
interest at G per cent V 1 

150 -^.OG =82500. An.«. ^ 

ExAMrr.r. — What is the present worth of a perpetuity of $150 
a year, payable in semi-annual instalments, and to commence 4 
months hence ; interest 7 per cent V 

r ' 12 

Example. — Required the present worth of a perpetuity of 
$400 a year, payable in quarterly payments, and to commence 6 
years hence ; interest and discount being 5 per cent., compound* 
per year. 

r) _ "^' ' " ^ ="" '^ " % = S6081.65. Ans. 



ni + ir) 400X1.''h^ 



The Amount, Time, and Rate given, to find the Annuity, 
When payable in a single payment yearly from the present time, 

P = (H^^ 5 balf-yearly, P = [-(^ ^.j-^'iKl +ir) ' 

third-yearly, P = ^^^ .,),-y^,)._ ^j ; quarterly, 
^ 

(i+.i'-)C(i + '-)--i] ' 



ANNUITIES. 163 

and so, pro rata, for other fractional units of the integral unit. 
Therefore (l 4- r)-^ -^ 1 = ^^ , or ^ , or ^ 

Ar 

Example. — What annuity, payable in quarterly payments 
/rom the present time, will amount to S3000 in 12 years; interest, 
being compound per annum, at 8 per cent. ? 

3000 X-08-f- [(108^^ — 1) XI. '-^V^]= 3153.48. Ans. 

Example. — What length of time must a current annuity of 
$400, payable in quarterly payments, remain unpaid, that it may 
amount to $2500 ; interest being 7 per cent, yearly ? 

2o00 X .07 ^ ,4263094 = 5 + years, and 5 years by table of 
400X1."-^' ^-^ ^ ^ 

(l+ry^l = .402552 : therefore ('-^^-^t- l) ~=oOS 
^ ' ^ \ .402od2 / .0^ 

days, 5 years, 308 days. Ans. 



The Py^esent Worth, Time, and Rate given, to find the Annuity, 
When payable in a single payment yearly from the present time, 

yearly, P = ^^^ ^,.y\_J^^[ ^ ,^^ ; quarter-yearly, P = 

\w+¥=^^¥Vy'''- Therefore, (l-|-0-;=p-:^ = 
P(l + jr) _ P(l + fr) 
P( 1 + ir) _ D. - P( 1 + f r) - D. ' "^^^ 

Example. — What annuity, payable in half-yearly instalments, 
, and to continue 3 years, is at present worth $1335.13 ; discount and 
interest being compound per year, at 7 per cent ? 

1335.13X.07X1. 07^ ^ ^^ , 

— „ , — 07- ~ $500. Ans. 

(1.07^— .1) Xl-^ 



164 ANNUITlEa. 



OF INSTALMENTS GENERALLY. 

Any certain sum of money to be paid on a debt periodically 
until the debt is paid is called an instalment ; and a debt so made 
payable is said to be payable by instalmcnis. 

Let D = principal or debt to be paid, 

" = number of years in which the debt is to be paid, 

r== rate of interest per annum, 

p = instalment or periodical payment 

When the instalments are payable yearly, and the debt is at 

interest, 

When payable half-yearly, 

_ Dr(l + r)' 

^'-2[(l+r)'-l](l-f ir]' 

/,|r\-- ^(^ + ^'-) + f . n- 2/>r(l+r)--l](l+jr) 
^ "*" ' "[;'(!+ iO+y^] -\>r'"- r(l -f- r)« 

When the debt is not on interest, and the instalments arc pay- 
able yearly, 

^-(l+r)-— 1 ' ^^ +'^ - p '^"- r 

Example. — What yearly instahiient will par a debt of S4000 
in 4 years, the debt being on interest the while, at 6 per cent 
annually ? 

4000 X .06 X 1.06* -i- (1.06* — 1) = S1154.37. Ans. 

Example. — AVhat semi-annual instalment will pay a debt of 
$4500 in 3 yeai-s, the debt bearing interest at 7 per cent, yearly ? 

4500 X .07 X 1.07* ^^.« ^« M 

^ ^ — S842.62. Ans. 



2X (1.07^ — 1) X 1.0175 



AKHUITIES. 165 

When a debt has been diminished at regular intervals by the 
payment of a constant sum, to find the remaining debt at the close 
of the last payment. 

When the debt is on inteixist, and the payments kave been made 
yearly from the date of the debt, 

d = ;>—(;? — DO(i +0\ /^ , ^Y ^ p_-2^ 

r ^ \ T ) ^ — j^ 

When tlie payments have been made half-yearly. 

Example. — On a debt of Si 000, drawing interest the while 
at 8 per cent a year, there has been paid yearly, from the date of 
the debt, $200 for 6 years: required the unpaid debt at the close 
of the last payment. 

[200 — 1.08^ (200 — ,08 X 1000)] -f- ,08 = $119.69. Ans. 

Example. — On a note of hand for Si 000, and interest from 
date, at 8 per cent annually, the following payments have been 
made; viz., Si 00 at the close of every half-year from the date of 
the note, for 6 years. How much remained unpaid at the close 
of the last payment ? 

[204 — 1,08^(204 •— .08 X 1000)] -^ .08 = $90,34, Ans. 



166 PERMUTATION, 



PERMUTATION. 

Permutation, in the mathematics, has reference ti the grerateel 
number of unlike rehitive positions, that a given number of thinga, 
either wholly unlike, or unlike only in jiart, may be placed in. It 
considers the nunilx'r of changes, therefore, that may be made, in 
the arrangement of the things, under different given circumstiinccs. 

To find the number of changes that can be made in the order of arrange- 
ment of a given nutnber of things, when the things are all different. 

RuLK. — Find the product of the natural series of numbers, from 
1 up to the given number of things, inclunive ; and that pnxluct will 
be the number of changes or permutations that may bo made. 

Example. — In liow many different relative positiomi may 12 per- 
sons bo seated at a table ? 

1 X 2 X 3 X 4 X '') X <> X 7 X S X '•> X in X H X 12 = 

47'J,001,000. Ans. 

To find the number of changes thai can be made in the order of arrange- 
ment of a given nuniJtcr of things, when that nmnbcr is composed of 
several diffcnmt things, and of several which arc alifce. 

Rule. — Find the number of changes that could be made if tho 
thinga were all unlike, as in first example. Then find the number 
of changes that CDuld l)e made with the several things of each kind, 
if they were unlike. I>ii8tly, divide tho number first found by the 
product of the numbers last found, and the quutient will l>o tho 
number of permutations or changes that the collection admits of. 

Example. — Required the numlxir of permutations that can bn 
made with the letters a, bh, ccc, dddd, = 10 letters. 

1 X 2 X 3 X 4 X •'» X G X 7 X 8 X X 1 = 30^800 _ 

1X2X0X24= 288 — 1-'^^^- ^'*-'- 

To find the number of prrnnitations that can />e mcule with a given num- 
ber of different things, by taking an assigned number of them at 
a time. 

Rule. — Take a series of numbers beginning with the numlxjr of 

things given, and decreashig by 1 continually, until the numl>or of 
terms is equal to the numl)er of things that are to be taken at a 
time ; then will tho product of the series l>o the number of changes 
that may bo made. 



CO]tfBINATION. 167 

Example. — What number of changes can be made with the 
numbers 1, 2, 3, 4, 5, 6, taking three of them at a time? 

6 — 1 = 5, 5 — 1 = 4, then 6X5X4 = 120. Ans, 
What number, by taking 4 of them at a time ? 
6X5x4X3 = 360. Ans, 

Example. — Arrange the three letters a, b, c, into the greatest 
number of permutations possible. 

abc, ad), hac, bca, cab, cba, = 6 permutations. Ans, 

Example. — Arrange the four letters €i, b, a, b, into the greatest 
number of permutations possible. 

abab, aabb^ abba, bbaa, baba, baab, = 6 permutations. Ans 



COMBINATION. 

Combination, in the mathematics, has reference to the number of 
unlike groups, which may be formed from a given number of differ- 
ent things, by taking any assigned number of them, less than the 
whole at a time. It does not regard the relative positions of the 
things, one with another, in any of the collections or groups. But 
it exacts tha^t each group, in all instances, shall have the assigned 
number of members in it, and that, in every group, in every instance, 
there shall be a like number of members. It exacts, therefore, thai 
no two groups shall be composed of precisely the same members. 

To find the number of combinations that can be made from a given 
number of different things, by taking any given nvjnber of them at a 
time, 

Rule, — Take a series of numbers beginning with that which is 
equal to the number of things from which the combinations are to 
be made, and decreasing by 1, continually, until the number of 
terms is equal to the number of things that are to be taken at a time, 
and find the product of those numbers or terms. Then take the natural 
series, 1, 2, 3, 4, &c., up to the number of things that are to be taken 
at a time, and find the product of that series. Lastly, divide the 
product first found by the product last found, and the quotient will 
express the number of combinations that can be made. 

Example. — What number of combinations can be made feom 8 
different things, by taking 4 of them at a time'? 
8X7X0X 5 _ 1680 
1X2X3X4 24 — '"• '^'**- 



50. Ans^ 



168 eo^TBmATiOK. 

What number, by taking 5 of them at a timo ? 

8X7X6X5X4 G720 

1X2X3X4X5- 120 
What number, by taking 3 of them at a time ? 

8X7X0^336 

1X2X3 IT "^ ^- ^^' 

Example. — What numlx^r of combinations can bo made from 5 
iQfferent things, by taking three of them at a time ? 

5X4X3 GO 

1 X 2 X 3 = = '^^- '^'^' 

What number, by taking 2 of them at a time ? 



1 X 



9. 9 



ExAirPLE. — Funn 5 letters, a, h, c, «/, «, into 10 eombioatioDS of 2 
letters each ; tliat i», into 10 unlike groups of two letters each. 

aJ, flc, mdy oe, be, bd, he, cd, ce, de. Ans, 

Form them into the greatest number of combinations possible* m 
collections of three eaoli. 

&bCy ahi, abcy acd, ace, odcy bed, bce^ bde, cde. Atu. 



PROBLEMS. 169 



PROBLEMS. 



pROB. I. — The sum and difference of two numbers given, to find the 
numbers. 

Let a = the greater number, 

b = the less number ; then — 

^+iZL^^==5,and^> + «^5 = a; or 
2 ' 

a -\- b -^^ a ^ b 

o = ^> BXid a — a ^j^b=ib. 

Prob. II. — The sum of two numbers and their product given, to find 
the numbers. 

/s/ {a-{'bY ■— {aXh X ^) =^ a ^b, and 

a'\-b — a^b 

^ = b, and b-^- a^b = a. 

Prob. III. — The difference of two numbers and their product giveHf 
to find the numbers, 

V (a ^ Z^)2 + (a X ^ X 4) = fl + 5, and 

a-\-b — a^b 

-^ = b, and b-\-a^b = a, 

Prob. IV. — The sum of two numbers and the sum of their squares 
given, to find the numbers. 

/v/(a'^-f-^2)x2— {a-^bf^au^b, thence, by Prob. I. 

Prob. V. — The difference of two numbers and the sum of their 
squares given, to find the numbers, 

a/(6z-^ + ^^) X2— («^/^)2 = « + Z>, thence, by Prob. I. 

Prob. VI. — The sum of two numbers and tlie difference of their 
squares given, to find the numbers, 

d^^lP' a-\-b — a^b 

— -r— r z=aj^b; rt = Z>; b'-\-a^b=-a; or 

{a-\-bY'—a^j^b'^ 
^^ipp^2 — =^and« + 5 — 5 = fl5. 

Prob. VII. — The difference of two numbers and the difference of 
their squxires given, to find the numbers, 

d^^b^ a'-\-h — a^b 

r=a-{-b: 7i =6; 6 + «s/)6 = a. 

15 



170 PROBLEMS. 

Prob. VUi. — The product of two numbers and the sum of thar 
squares given, to find the numbers. 



V(a--h^ — fl X /^ X 2) = fl^ ^>, and 

V(a2 + 52^aX^X2) = a4-i, and 

a-{-b — fl^ b 

o = b, and b-\-aj^ b^^a, 

Prob. LX. — T?ie sum of two numbers^ and the product of those 
numbers plus the square of one of the numbers, in another sum given f 
to find the numbers, 

aXb + b!^ 

a-\-b ' 

Prob. X. — The product of two numiMrs mid the relation of those 

numbers to each other givrn, to find the numbers. 

/f oXh\ 
\ r ye r^ l Xr = a, OT X^=;f>; r Ix^ng the term in rclaiioo 

representing; t!ic ^i^rcator number, and r^ Injing the temi in n*latioa 

representing the Ichs. 

Prob. XT. — The sum of the Sijunrrs of two numl>crs, and the reUh 
tion of those numlxrs to each other given , to find fhr nninltcrs. 






Prob. XII. — Thr sum of three num})ers xrhirh are in arithmrtical 
progression, and the sum of their S(/uares given, to find the nut fibers. 

Let a = the greatest numlxjr, 
b = the middle number, 
c = the least number ; then 

a + ^ + r 

q = /', and a-f-/y-)-c — iaafl-J-c. ^ 

V(«- + <^')X2— (a4-r)2 = a^c. 

a-\-c — a lt c 

}y = c, and c -\- a ^ c = a. 

Example. — The sum of three numbers which are in arithmetical 
progression is 18, and the sum of their squares is 140 ; what aro 
the numbers ? 



PROBLEMS. 171 

18 -7- 3 = 6 = 5, and 18 — 6 = 12 = sum of a and c. 

140 — 62 = 104 = sum of a^ and c^, 

V (104 X 2 — 12') = 8 = fl — c. 

12 — 8 = 4 -r- 2 = 2 = c, and 2 + 8 = 10 = a; the numbers, 

therefore, are 2, 6, and 10. Ans. 

Note. — Half the sum of the first and third of three numbers forming an arithmetical 
progression is equal to the second number. 

Prob. XIII. — The sum of three numbers which are in arithmetical 
progression added to the sum of the greatest and least ^ and the sum of 
the squares of the numbers given ^ to find the numbers. 

flf + ^ + C + fl-f c 

^ = ^) and 

fl-j-^-j-c-j-^ + c — h 

o = a -|- c ; thence, by Prob. XU. 

Prob. XIV. — The sum of three numbers which are in arithmetical 
progression added to the sum of twice the greatest and twice the least, 
and the sum of the squares of the numbers given, to find the-numbers, 

Sa-\~3c-\-b 
^ = b, and 

3a + 5 + 3c— 5 

o = a -|- c ; thence, by Prob. XU. 

To find the altitude of an equilateral four-sided pyramid, the slant 
height and side of the base being known, 

'V (S^ — 4 A^) = A ; S being the slant height, A a side of the base, 

and h the altitude. 

/\/(F2 — (4 A)- X 2) = A ; F being the linear edge. 
V (F^ — f) =h ; f being half the diagonal. 
a/(F2 — AX.707lV = A; 

To find the altitude of the frustum of an equilateral rectangular 
pyramid, 

1/ S^ — I — 9 — j = /i ; S being the slant height, A a side of the 
greater base, and a a side of the less. 

1/ F2— ( — -— \y^2 = h; F being the slant height meas- 
ured along an angle. 

* Diameter X '"^OTl = side of inscribed square. See Centres of Surfaces. 



SECTION IV. 

GEOMETRY, — PRACTICAL AND ILLUSTRATIVE. 



Geometry is the science that treats of the properties of fipind 
space. It is the science of mafcnituJe in general, and coniprcli'ii !i 
the mensuration of solids, surfaces, lines, and their various r- hi- 
tions. 



DEFINITIONS. 

A Point has position, but not ma^niludc. 

A Line is hMi^^ih without l)readlh, and is cither Right, Curved, or 
Miral. When no particiihir line is sixjcified, a right line is meant. 

A Right Line is a straight line, or the shortest distance between 
two points. 

A Mixed Line Mi a rif^ht line and curved line united. 

Lines are parallel, oblitpic, perpendicular, or tangcntical, one to an- 
other. 

An Area, surface, s^tprrfirirs, is the space contained within the out- 
line or perimeter of a figure; it h:is no thickness, and is estimated 
in the S(pmrc of some unit of measure, as square inch, st^uare 
ijard, See. 

A Solid has length, breadth, and thickness, and its contents aro 
estimated in the n//>c of some unit of measure. 

An Angle is the diverging of two lines from each other, and is 
right, acute, or obtuse. 

A RigJil Angle has one line perpendicular to another and resting 
upon it. 

A Triangle, or trigon, is a figure having three sides. 

An E<piilatcral Triangle has all its sides equal.* 

An Isosceles Triangle has two of its sides equal. 

A Scalene Triangle has no two sides equal. 

A Right-angled Triangle has one right angle. 

An Obtuse-angled Triangle has one obtuse angle. 

An Acute-angled Triangle has all its angles acute. 



GEOMETRY. 



17a 



A Quadrangle^ tetragon, quadrilateral, is a figure having four 
sides. 

A Parallelogram is a quadrilateral figure whose opposite sides are 
parallel and equal. 

A Rectangle is a parallelogram whose opposite sides are equal, its 
angles right angles, and its length greater than its breadth. 

A Square is an equilateral rectangle. 

A Rhomboid is a quadrilateral, having its opposite sides equal and 
parallel, its angles oblique, and a length greater than its breadth. 

A Rhombus, or lozenge, is an equilateral four-sided figure, having 
oblique angles. 

A Trapezium is a quadrilateral having no two sides parallel. 

A Trapezoid is any four-sided figure having two of its sides paral- 
lel, but of unequal lengths. 

A Diagonal is a line joining any two opposite angles of a figure 
having four or more sides. 

A Polygon is a plain figure having more than four sides. 

A Regular Polygon has all its sides equal. 

An Irregular Polygon has not all its sides equal. 

A Pentagon has five sides ; a hexagon, six ; a heptagon, seven ; 
an octagon, eight ; a nonagon, nine ; a decagon, ten ; an undecagon^ 
eleven ; a dodecagon, twelve. 

The Perimeter of a figure is its bounds, limits, or outline. It is to 
other figures what the circumference is to the circle, and the perimeter 
of any portion of a figure is the outline of that portion. 

The Altitude, or height, of a figure, is a perpendicular let fall from 
its vertex, or highest point, to the opposite side or end, its base. 

The Base of a triangle is that side that is placed parallel to the 
horizon ; and of figures in general the base is that end, or side, upon 
which the figure is supposed to stand or rest. The sides of a triangle 
are often called the legs. In a right-angled triangle, the longest side, 
or line which subtends the right angle, is called the hypotenuse, and 
of the other two sides, one is the base, and the other the perpendic- 
ular. 



A Circle is a plane figure, bounded 
by a curve line, called the circumfer- 
ence or periphery, every part of which 
is equi-distant from a point within called 
the centre, as A C B D, in the diagram. 
The circumference itself is of^en called 
a circle. 

The Radius — semi-diameter — is a 
line drawn from the centre to the cir- 
cumference, as A, or C. 

The Diameter is a line drawn from 
the circumference through the centre to the opposite side, as A 6. 
15* 




174 GEOMETKY. 

A Semicircle is half a circle, or it is half the circumference of a 
circle, as A C B. 

A Quadrant is a quarter of a circle. It is also sometimes a quar- 
ter of the circumference, as A C. 

An Arc is any portion of the circumference, as B c a, or ^ C ^. 

A Chord, or suhtensey is a right line joining the extremities of aa 
arc, as B a, or h //. 

A Segment is the portion of a circle contained between the arc 
and its chord, as the space Ix^tween the arc B c a and its chord Ba, 
or between the arc h I) 7, or h C 7, and the dionl h g. 

A Sector is the space between two radii, or lines passing from the 
centre to the circumference, as the space B O a. 

A Secant is a line that cuts another line. In trigonometry, the 
secant of an arc is a right line drawn from the centre of a circle 
through one end of the arc, and terminated by a tangent drawn 
thruiiizh the other end ; thus, the secant of the arc B c a is the 
line 6 b. 

A Cosecant is the secant of the complement of an arc, as O 6. 

A Sine of an arc is a line drawn from one end of the arc per- 
pendicular to a radius drawn through the other end, as a ^, and is 
always eciual to half the chord of double the arc ; and the sine of 
an angle is the sine of the arc that measures that angle. 

The Versed Sine is that ix>rtion or part of the radius lying be- 
tween the foot of the sine and origin of the arc, as e B, and the 
versed sine of half the arc is that portion of the radius lying be- 
tween the chord and the arc, bisecting or dividing both at their 
centres, as k I), in the arc h I) g. It is the height of the arc or 
segment. 

The Cosine of an arc, or ani^le, is that portion of the radius lying 
between the sine and the centre, as c O. 

The Covcrscd Sine is the sine of the complement of an arc, or an- 
gle, or the coversed sine of the given arc, or angle ; thus, the line 
a d is the coversed sine of the arc B c a, or of the angle B O a. 

A Tanfrenf is a right line that touches a curve, and which, if pro* 
duced, will not cut it, — the tanirent of the arc B c n, is B b. 

A Cotant^rni is the tangent of the complement of an arc, or the tan- 
gent of an arc which is the complement of another arc to ninety 
degrees ; thus, the cotanf;ent to the arc B c a, is the line C b. 

The Complement is whrit remains of the quadrant of a circle, afler 
the angle has been taken therefrom, — the complement of the arc B 
c a, is rt C. 

The Siipplcnienf is what remains of a semi-circle aAer taking an 
angle therefrom, — the supj)lcmcnl of the arc B c rz, is a C A. 

A Gnomon is the space included between two similar parallelo- 
grams, one inscribed within the otlier, and having one angle common 
to them both. 



GEOMETRY. ' 175 

A Zonp. is the space between two parallel chords of a circle, — the 
space included between the lines A B and li g. 

A Lune, or Crescent^ is the space contained between the inter- 
sectinp: arcs of two eccentric circles, as i n s* 

A Circular Ring is the space between the circumferences of two 
concentric circles. 

An Oval is a figure of an elliptical form made up of arcs of 
circles. 

A Helix is a coil or spiral, or it is part of a spiral line. 

An Ogee^ cym.a^ or talon, is two circle arcs that tangent each 
other, and meet two parallel lines, either tangentical to the lines or 
at right-angles to the lines in given points. 



A Prism is a solid whose bases or ends are any similar, equal plane 
figures, and whose sides are parallelograms. 

A Parallelopiped is a solid having six sides, its angles right-angles, 
and its opposite sides equal. It is a prism, therefore, whose base is 
a parallelogram. 

A Cube is a solid having six equal sides and all its angles right- 
angles. It is a square prism. 

A Pri^moid is a solid whose bases are parallel but unequal, and 
whose sides are quadrilateral. 

A Pyramid is a solid having any plane rectilinear figure for its 
base, and all its sides, more or less, terminating in a point, called 
its vertex or summit. 

A Cylinder is a circular solid, having a uniform diameter, and 
equal and parallel circles for its ends. 

A Cone is a solid having a circle for its base, and a true taper 
therefrom to its vertex. 

Conic Sections are the lines formed by the intersections of a 
plane with a conic surface. They are the triangle^ circle^ ellipse^ 
parabola and hyperbola, 

A Conoid h a solid generated by the revolving of a parabola, or 
hyperbola around its axis. 

A Spheroid is a solid generated by the revolving of an ellipse 
about either of its axes or diameters. 

The Transverse or Major axis of an ellipse is its longest diameter, 
or the distance, lengthwise, through its centre. 

The Conjugate or Minor axis of an ellipse is the shorter of the 
two diameters, — a right line bisecting the transverse. If the gen- 
erating ellipse revolves about its major axis, the spheroid is prolate, 
or oblong ; if about its minor axis, it is oblate^ or flattened. 

An Ordinate is a right line drawn from any point of the curve of 
a conic section to either of its diameters, and perpendicular to that 
diameter. 



176 GEOMETRY. 

The AbsctsstB of a conic section are the parts of either diameter, or 
axis, lying between their respective vertices and an ordinate*. 

The Parameter — latus rectum o( n parabola — is a third propor- 
tional to any diameter and its conjugate. In the parabola it is a third 
proportional to any abscissa and its ordinate, or to the altitude of 
the figure and half the base. 

The Focus is the point in the axis where the ordinate is equal to 
half the parameter. 

A Sphere, or globe, is a perfectly round substance — a solid con- 
tained under a curved surface, every point of which is equally distant 
from a point within, called the centre. Its axis, or diameter, is any 
right line passin^j from a side through the centre to the opposite side. 
A hemisphere is half a spliere. 

A Frustum of any solid figure, as of a cone, pyramid, etc., is the 
part remaining after a segment has been cut off. 

An Ungula is the section of a cylinder cut off by a plane obliqno 
to the base. 

The Slant height of a regular figure is the length of one of its 
sides, or the distance from the outline of its base to its vertex, or 
summit. 

To bisect a right linej A B, 6y a perpendicular. 

Set one foot of the dividers in A, and with the other extended 80 
as to reach somewhat beyond the middle of the line, describe arcs 
above and below the line ; then, with one foot of the dividers in B, 
describe arcs crossing the former ; a line drawn from the intersection 
of the arcs above the line to the intersection of those below, will divide 
the line into two equal parts. 
To erect a perpendicular on a git'e?i jwint in a 

straight line, or to draw a line at a right 

angle to another line. 

Set one foot of the dividers in the given 
point, f, and with the other extended to any 
convenient distance, as to A, mark ecpial dis- 
tances on each side c, as c A, c B ; and from 
A and B as centres, with the dividers ex- 
tended to a distance somewhat greater* than 
that between c and A, or c and B, describe 
arcs cutting each other above the line, as at 
d; a line drawn from the intersection of the arcs, d, to the point c, 
will be perpendicular to the line A B, or will form a right angle with 
the line c A, or c B. 

From a point, d, to let fall a line perpendicular to another Une, A B. 
Set one foot of the dividers in d, and with the other extended so as 
to reach beyond the line A B, describe an arc cutting the line A Bf 




GEOMETRY. 



177 




in e and n; then with one foot of the dividers in e, and the other ex- 
tended to morp than half the distance between e and n, describe the 
arc g; then with one foot of the dividers in n, describe an arc cutting 
the arc ^ in ^ ; a line drawn from the point d through c to the inter- 
section of the arcs at g, will be the perpendicular required. 
To erect a perpendicular upon the end of a line, as at c, on the line A c, 

Set one foot of the dividers in c, and, at any convenient radius, de- 
scribe the arc e h k; with one foot of the dividers in e, cut the arc in 
h, and with one foot in A, cut it in k; from A as a centre, and ^ as a 
centre, describe arcs cutting each other dX d; a line drawn from the 
intersection of the arcs, d, to the point c, will be perpendicular to the 
line A c. 

To draw a circle through any three given points not in a straight line, 
and to find the centre or radius of a circle, or arc. 

Let the given points be a, b, c. With the dividers opened to any 
convenient distance, and either point the centre, 
(as h,) describe the portion of a circle r s t, 
and with the same radius and a the centre, 
describe an arc cutting r 5 ^ in 7' and 5, 
and with the same radius and c the centre, 
describe an arc cutting r 5 ^ in ^ and s ; 
draw lines through the points where the arcs 
cut each other, (the lines r 5 and t s,) and their 
point of union will indicate the centre of the 
circle or point, from which as a centre a circle, if drawn, will pass 
through the three given points. 

To find the length of an arc of a circle. 

Take i the length of the chord of the arc, (A B,) and with the 
dividers at that radius, and A the centre, 
cut the arc in c; also, with the dividers 
at the same radius and B the centre, cut 
the chord in d; draw the line c d, and 
twice its length is the length of the arc, nearly. 
From a given point, to draw a tangent to a 
circle. 

Let A represent the given point, and C, the 
centre of the circle. Draw a line from A to 
C ; bisect the line, and with the point of bisec- 
tion as centre, describe the semicircle ADC; 
then draw a right line, A B, cutting the semi- 
circle at the point where it intersects the circle, 
which is the tangent sought. 
To draw from or to the circumference of a circle, lines tending to the 
centre of said circle, when the centre is inaccessible. 

Divide the circle, or such portion thereof as required, into the 





17 



GEOMETRY. 




desired number of equal parts, and designate the points of division, upoa 

or at an uniform distance from the pe- 
riphery ; then, with any radius less 
than two of the parts, and the re- 
spective points as centres, describe 
arcs cutting each other, as A 1, r/ 1 ; 
c 2, e 2, &c. ; draw the lines c 1, </ 2, 
e 3, &c., which lend to the centre, as required. 

When a portion of a drcle or segmrnt only is ttscd^ to draw the end lines, 

A 5, 13 s, 

I^roducc the arc each way beyond A an»l B to a distance equal to 
that between A and c, and with the extromrs of the extensions as 
centres, and the second points inward ihcrofroni as centres, de.scril)e 
the inters<'ciinpf arc^, s s. Or, with c as a centre, describe the arc 5, 
and with the radius c 1, and A or B as the centre, describe the inter- 
secting arcs ; lines from the intersection of the arcs, s 5, to their 
respective points, A and B, will tend to the centre, as required. 

To describe an elliptic arch on a given conjugate diameter. 

Let A B be the given diameter, which bisect, and from the |Muiit 
of bisection, r, erect the peri>en(licular 
e/i equal in length, or proportidnal in 
length to the height of the intended 
arch ; make c a, c b^ cacii vi\\ii\\ to cf, 
and bis(M't rfm o; draw the lines ao c 
and b o r/, and with the radius a B or 
b A, and a and b as centres, descriln? the 
arcs A</ and B c; then, with the radius 
o d, or c, and o as the centre, describe the arc dfc, and the arch is 
completed. 

To dcscriftc an Ellipse of given length and breadth. 
Let the line A B equal the given length, or transverse diameter, 
and tlie line C D the conjugate, and let these lines bisect each other, 
forming right angles, on either side, as 
at c. Lay otf tluj distance C D on the 
line A B, as from A to /, and divide 
the distance / B into three equal parts. 
Fn)m r, on the line A 15, set oil' two of 
the parts each way, ^s e a, c h; and 
from ^, or A, designate the distance a h 
on the line C D, as at i and n ; from i 
tlraw the lines i / and i r, and from n, 
the lines 71 5 and n A-, passing through 
the points a and h and cutting each other 





GEOMETRY. 



179 




therein. From the point n, as a centre, describe the arc s k, and 
from i, as a centre, the arc r t ; also, from a, as a centre, describe 
the arc t s, and from A, as a centre, the arc/^ r, and the required ellipse 
is drawn. 

Note. — An architrave of any depth desired, may be readily described on the above. 

To construct an arc or segment of a circle of large radius. 
Draw the chord A B equal in length, or proportional in length, to 
the chord of the arc intended ; also draw E F parallel to the chord and 
at a distance therefrom equal to the 
height of the intended segment. 
Bisect the chord, and from the 
point of bisection erect the perpen- 
dicular C D ; draw the right lines 
A D, D B, and draw A E at a right 
angle to A D, and B F at a right angle to B D, and erect the perpen- 
diculars A y, B/. Divide A B into any even number of equal parts, 
and divide E F into the same number of equal parts, and draw the 
lines 1, 1 ; 2, 2 ; 3, 3, &c. Divide A/, B/, each into half the 
number of equal parts the chord A B is divided into, and draw lines 
from D to the points of division, respectively. A curve passing 
through the intersections of the crossing Imes bearing the same num- 
ber will describe the arc required. 

To describe an elliptic arch, the span and rise being given. 
Bisect the given chord, or span, A B, with a line at right angles 
therewith, and let the portion of the line C D be the rise intended. 
Erect A a equal and parallel to C D, 
and draw the line a D equal and parallel 
to A C. Bisect A a in t, and draw the 
line t D ; make C b equal to C D, and draw 
the line a b. Bisect the portion of ^ D 
lying between the line a b and the point D, 
and from the point of bisection, at right 
angles to ^ D, draw the line meeting D C 
in g ; then draw the line a g. Let/ des- 
ignate the point where the line a g cuts 
» A B, and make A B equal to A/, and draw 
the line g h e. With the radius g D, and ^ as a centre, describe the 
arc from the line a g io e, and with the radius A /, and /and h as 
centres, describe the arcs A s, e B, which will complete the arch 
required. 




180 



GEOMETRY. 




To draw a Gothic arch. 

Fig. 1. — With the chord A B as radius, and A and B as centres, 
describe the arcs A r and H c. 

Fig. 2. — Divide the given chord, A B, into three equal parts, and 
witli two of the parts as radius, and b and c as centres, describe the 
arcs A d^ B d. 

Fig. 3. — Divide the pivfcn chord, A B, into three equal parts, « 
and/, and from the points A and B let fall the perpendiculars A a, 
B h, equal in lonijlh to two of the divisions of the chord. Draw the 
lines a h and h g, i):issinf,' through the divisions e and/", and with one 
of the divisions as radius, and c and / as centres, describe the arcs 
A i,^ B h ; also, with the radius a hy or g by and a and b as centres, 
describe the arcs g v and v h. 

Fig. 4. — Divide the given chord, A B, 
into three equal parts, a and b, and with the 
radius two of the parts, and A,a,/>, and B as 
centres, describe the four arcs b c , B </, a d, 
A c ; then, with the radius one of the parts, 
and a and l> as centres, describe the arcs A h 
and B g; then, with the radius c g, or d A, 
and c and d as centres, describe the arcs h t 
and g I. 




GEOMETRY. 



181 




To describe a Regular Polygon of any numher of sides not exceeding 
tvjelve, to a given face or chord line. 
Let A B be the given face, which bisect, 
and from the point of bisection, and at aright 
angle to A B, draw the line C D. With the 
radius A B, describe the arc B 6, and divide 
the arc into six equal parts, and from 6 con- 
tinue the divisions on the line C D, as v, 7, 
8, 9, (fee, to 12. A circle whose radius is 
B V, B 6, B 7, &c., will contain the given 
face or side (A B) 5, 6, 7, &c., times. 

To inscribe any Regular Polygon in a given circle^ or to a given 

diameter. 

Let A B be the given diameter, which divide 
into as many equal parts as the polygon is to 
have sides; then, with the radius A B, and A 
and B as centres, describe the arcs cutting each 
other as at C. From the intersection of the 
arcs at C, draw a line through the second point 
of division on the diameter to the peripliery, as 
C D, and the cliord of the arc, D A, will be one 
side of the polygon required nearly. 

Let a pentagon be required. See Fig. ' 

To circumscribe a Regular Polygon of any 
given numher of sides ^ about a given circle. 
Divide the given circle into as many equal 
parts as the polygon is to have sides, and de- 
fine the points of division on the circle ; then 
draw lines from the centre, o, to each of the 
respective points, as o a, o /;, &c. Through 
these points, and at a right angle to the line 
leading from the centre thereto, draw the 
lines A B, B C, &c., which will complete the figure. 

Let it be required to circumscribe with a pentagon ; — see Fig, 

To construct a square whose area shall he that of a given triangle. 
Let A B C be the given triangle. 

From B let fall the perpendicular B <2; 
make A b equal to half B a ; bisect b C, and 
from the point of bisection as a centre, de- 
scribe the semicircle ^<ZC : erect the perpen- 
dicular, A d, which will be the side of the 
6(iuare required. 

16 





182 



GEOMETRY. 



1 1 1 aD 5 • r 




To construct a Parabola, 
Let A B equal the base, which bisect, 
and upon the point of bisection erect the 
perpendicular C I), the altitude; Icltlie 
line E F be half the length of A B, and 
let it lie parallel to A B, and at the dis- 
tance D C from the base, and let it be bi- 
sected by C D, as at 1). Draw the lines 
E A, F B, and divide them, together with 
E D, D F, into four or any numl>er of 
equal parts, as 1 1,2 2, <&c., and draw the 

lines connectinir the respective points of ^^- 

division. The inner intersections of the "^ 

said lines with each other define the curve of a parabola. 

To rnnstntcta Hyperbola, 

Ijot A B equal the longest or transverse diameter, and C D, pei^ 
pendicular to it, the 
conjugate, and let iho 
line A B be produced or 
extended from its re- 
spective limits each 
way, as to «, r, r, &c. 
Bisect A B in o, and 
with the radius o C, or 
o D, and as the centre, a\- 
describe the circle C c 
D a. Divide A B pro- 
duced from B, into any 
number of parts, as r,ry 
r, &c., and with the ra- 
dii A r and B r, and the 
foci a and c as centres, 
describe arcs cutting 
each other as in 5 5, &c. The intersections of the arcs with each other 
will (l.finp the curve of the hyperlwla. 




To bisect any given triangle. 
Let A B C be the given triangle. 
Bisect one of the legs, as A B, and with the 
point of bisection as a centre describe the semicir- 
cle, B fl A ; bisect the semicircle, as at a, and with 
the radius B <7, and B as a centre, describe the 
are a h : from // erect the line b c paralk'l to A (', 
which will bisect the given triangle, or divide it 
into two equal parts, as required. 




GEOMETRY. 



183 




To draw an equilateral triangle ivhose area shall 
be that of two given equilateral triangles. 
Let the given triangles be A and B. 
Draw a line D E equal in length to one side of 
the larger triangle, and upon one end thereof, and 
at a right angle therewith, erect a line equal in 
length to one side of the less triangle, as D C ; 
then draw the line C E, which will be one leg of 
the triangle required. The equilateral triangle 
C E F contains an area equal to the triangles A 
and B. 

Note. — The same process is applicable to rectangular figures. 



To draw a circle ivhose area shall be that of two given circles. 
Let the circles A and B be the given 
circles. Draw a line whose length shall 
be equal to the diameter of the larger 
circle, and upon the end thereof erect a 
perpendicular equal in length to the di- 
ameter of the lesser circle, as C D E ; 
draw the line E C, and bisect it in z, and 
with the radius i C, or i E, and i as the 
centre, describe the circle E D F, whose 
area will be that of the two ijiven circles. 




To construct a toothed or cog wheel. 

Divide the pitch circle, a a, into as m.any equal parts as there are 
to be teeth or cogs; then, 
with the dividers extended to 
\\ tinies one of those parts, 
and the point 5 as a centre, 
describe the arcs i v, v i; 
then, with the same radius, 
and r and t as centres, de- 
scribe the arcs v/and/r, so 
continuing until the upper 
sections of all the cogs are defined. The lower sections are bounded 
by straight lines tending to the centre of the wheel. See Teeth of 
Wheels. 

Note. — The pitch of a wheel is the rectilinear distance from the centre of one cog to 
the centre of the next contiguous, measured upon the pitch circle ; and that portion of 
the length of a tooth lying between the lines a and b is usually made equal to i the pitch ; 
end that portion lying between the lines a and c is usually made equal to \ the pitch. 




184 CONIC SECTIONS. 



OF THE CONIC SECTIONS. 

The conic sections are the elements of geometrj'. They 
arc Hues, and nothing more. The doctrine of their* relations' is the 
sciKNCK of geometry. Geomfthy is lines in ])Osition. 

The conic sections are tlie lines formed by the intersections of a 
plane with a conic surface. They are the triangle^ circle^ ellipse^ 
parabola, and hyperbola. With a single exception, they are curved 
lines or curves. 

To the conic sections In'long the focu parameters^ ohschsfFy and 
orc/inatesy which are explained in a general way under defini- 
tions, p. 175, 176. 

The locusy or place, of a conic seetion is determined by a point 
called till' generatrix^ which is supposed to move in accordance with 
the law of the line. 

The equation of a line expresses the relation between the ordi- 
nate and abscissa of every point in the line, or between the co-ordi- 
nates of ever)' jwint in the line. 

The parameter is a double ordinate, and passes through the focal 
point. 

'J'he summit of an ordinate is a point in the locus of a line, in 
the }nilJt of the generatrix. Thus by the ])lotting of ordinates or 
double ordinates to dilForent abscissa*, the line is defined ; and, by 
properly connecting the points, it is practically constructed or 
Ibrmed. 

The eccentricitg of a conic section is its deviation from the centre. 

The (LxU of ahscissce is the axis, or diameter, in which the abscissa 
are taken, or the axis, or diameter, that is divi(le<l into abscissa*. 

The axis of ordinates is the axis, or diameter, that is parallel to 
the ordinates. 

The origin of a conic section is in the summit of the axis of 
abscisste. 

The axis of abscissa} proper is the major axis in any conic 
section. 

The asymptote is a right line that continually approaches a curve, 
but never meets it, however far both may be extended (sec CoN- 
STKUCTiONS, Hyperbola, p. 182). 

In all the conic sections, the parameter is a third proportional to 
any diameter and its conjugate. 

The radius vector^ or veetor, is a right line joining the centre of 
the sun to the centre of a planet. It is an element in astronomy, 
and has one of its extremities in the focus. 



LINES A'ND SUPERFICES. 185 



LONGIMETRY ANP PLANBIETRY; 

OR, 

LINES AND SUPERFICIES. 



TRIAXGLES. 

A TRIANGLE is the Simplest form in geometry, and the most Im- 
portant. It is a plane, three-sided, rectilinear figure, or is made up 
of three straight lines. It is the measure chiefly of itself, and is 
the measure of almost every geometrical form or structure. It is 
the measure of the resultant of forces acting from different direc- 
tions upon the same body. Similar triangles are measures of each 
other. 

It is generated, when a plane intersects both sides of a cone, 
from its apex along the plane of its axis. 

As a CONIC SECTION, it is simply two diverging lines having their 
origin in the same point, and subtended by an ordinate. 

Its equation is expressed by x \ y \' x' \ y' ; x being any abscissa 
on either leg reckoned from the angle opposite the ordinate, y the 
ordinate, x^ any other abscissa on the same leg reckoned from the 
same angle, y' ordinate to abscissa x^ 

In TRIGONOMETRY, it is a figure having three sides and three 
angles, and is supposed to have one of its angles in the centre of a 
circle, of which one of the sides is radius. 

In GEOMETRY, it is a figure having three sides, three angles, an 
apothegm, or perpendicular, and an area. 

Under its most favorable form (equilateral), it contains less area 
than a square of the same length of perimeter by 23 per cent. ; 
and less than a circle of the same length of perimeter by nearly 
40 per cent. 

16* 



186 



LINES AND SUPERnCIlS. 




RigJiUangled Triangle : — ADC, diagram. 

The longest side of this figure, 
A C, is usually called the hy- 
potenuse, and the other two 
sides, A D and D C, are called 
the sides or legs. The legs are 
perpendicular one to tlie other, 
and form the right-angle, D, or 
an^le of 90^ 

If one of the lees bo made 
bai^o, the other will ne perpendicular, and will be the altitude of the 
figure ; for the altitude of any triangle is a perpendicular, dropped 
from the vertical angle to the opposite side or base, and cither side 
may 1)0 made base ; thus, if A I) be made base, I) C will Ix? perjx'ndic- 
ular, and will be the altitud«» of the figure, and if I) C l)o made baae, 
A I) will l)e fn'q^t'ndicular, and will Ik; the altitude of the figure. 

If the hypotenuse be made base, the legs, A I) and D C, will still 
be ]MT|>enclicular one to the other, but the altitude of the figure, I) K 
or K I), a per[)endicular to tlie hy]Kitcnuse, will not Ik? shown. 

AV'hether the legs have equal lengths or unequal, so far as regards 
the principles of tlie figure, b immaterial. 



V(a1? + DC-)=AC; V(ACf — DC*): 
V(TT^'^— aIT) =DC. 



AD; 



jY D^ -i- A C = A E ^ Converting the right-angled trian- 

/ » i — o > gle A 1) C into two right-angled tri- 

V (A D" — A K) = E D. > . angles, A E D and E D, E D a 
leg conunon to both, and perpendicular to A C , and the altitude of 
the triangle A D C, therefore, A C being base. 



DC 


-7- A C = C E. 


A C X A E = A D^ 




V(i)c'- 


- C E-) = E D. 


A C X C E = D C". 




AD-' 
DC 
ED- 


-^ A E = A C. 
-T- C E = A C. 
-^ A E = C E. 


A E X C E = E D-. 
A d' : A c" : : A E : 
D C* : A D* : : C E : 


AC 
AE, 



D C" : A C : : C E : A C. 

Twice the area of a right-angled triangle, divided by the hypo- 
tenuse, is equal the pcipendicular to tlie liyix)tenuso, and tiic 
respective triangle will thereby be divided into two right-angled 
trianfiflcs having a side common to both.^ 



LINES AND SUPERriCi:^S. 



187 



Half the product of the two legs of a right-angled triangle equals 
the area of that triangle. 

2 area 2 area 

perp. to hypot. = hypotenuse. -^^-[^ = required leg. 

Of Oblique-angled Triangles, 

Every triangle not a right-angled triangle is either an acute- 
angled triangle or an obtuse-angled triangle, and these two (the 
acute-angled and obtuse-angled) are classed under the general name 
oblique-angled. The following principles are alike applicable to 
either. 

Let A B C be the triangle. 




AO^ 



BC^ 



2AB +iAB = AD, 

distance along the base, from 
the angle formed by the base 
and longest vertical side, at 
which a perpendicular dropped 
from the vertical angle will fall ; 
^B and 

a/ (AC^ — AId^) = D C, the 
perpendicular alluded to ; thus dividing the obtuse-angled triangle 
ABO into two right-angled triangles, ADC and B D C, D a leg 
common to both. 

Or, a/ ( A -f A D X AC — AD) = D ; for the sum of any 
two quantities multiplied by their difference is equal to the differ- 
ence of their squares. 



AB -^^^ j^^-^Q^^g^ andA/(AB2 — Bg-2)«=A^, 
A h, and ^(A^^ — A1^) = B A, per- 



2BC 

perpendicular to B produced 



AB 



BC' 



AC 

2 



2AC 
pendicular to A produced 



AC=AB+BC— 2ABKBD 

2 2 3 

g-jP__ AB-f BC —AC 

'~~ 2AB 



188 LINES AND 6UPERFICIKB. 

A n— ACjf AB' — Fc 
^^— 2(AB) 

AC — A^ = c7^,an(lC/7 + B Q = \^g; Ag +C^=AC. 

Twice the area of any triangle divided by the base is equal the 
perpendicular to the base. 

Ilalf the side of any triangle multiplied by the perpendicular 
to that side is equal the arra. 

of '^^^^:^x^:^:^ti^^^ '^^^^^ ^°- *»^»'- 

M /."f "h ^■'''."'^^••'•«' trian/?!,. a iMriu'ncllruI.ir dropped from either an-le will 
bisect thr sM ..opiM)Hte; and the tiianple will tl Jreby bf divided into two 
equal and sinnlar rlglit-anKkd scal« lu- trianplrs. u. . nu inio iwo 

In an isosceles trinn^rle, a peri)endieiihir dropped from the nnc'e Included 
by the equal Hiden will bisect tlu- Mde opposite. rV^! thr tH: pirwIU ?h bv 



, - . .. v^ < <|ij<ii Hint riiiiiiai riLTiii t!iani;IrM « 

equal and similar ri-ht-auKled iHoseeles tr i from n',, 

opposite one of the rqual gldes, it will fall < ;. , if tlir M 

be obtuse ; inside, if it be acute. ni....^.^ 

RECAPITULATIONS. 
To find (he Perpendicular of an Oblique-angled Scalene Triangle, 
the Sides being given. 
Lot / represent the longest side, s the shortest side, m the inter- 
mediate side, and h the perpendicular. 
When I is made base (B), 






When s is made base^ 

h 

WJien m is made base^ 



To find the Area of a Triangle. 

Rule. — ;Multiply the base by half the perpendicular height, or, 
the perpendicular height by half the base; or, multiply tlfe base 
by the altitude, and divide the product by 2, and the quotient will 
be the area. 

Example. —The base A C, of the triangle A CB, 
is 12 feet, and the altitude, D B, is 4 feet and G inches ; 
required the area. 

4.5 X ^^= 27 square feet. Ans. aw u 

To find the Area of a Triangle bg Means of the Sides. 
Rule. — Add the three sides together, and from half the sum sub- 
tract each side separately ; multiply the half sum and the three re- 




LINES AND SUPERFICIES, 189 

inainders into each other, and from the product extract the square 
root, which will be the area sought. 

Example. — The sides of a triangle are 30, 40, and GO rods; 
what area has the triangle ? 

30 + 40-1- 60 + 2 == G5 = i- sum of the three sides. 
Go — 30 1= 35, first remainder. 
G5 — 40 1= 25, second remainder. 
Go — 60 1= 5, third remainder. 
65 X 35 X 25 X 5 = v/2843 75 = 533. 2G square rods. Ans. 

To find the hypotenuse of a triangle^ the other two sides being given. 
Rule. — Add the scjuare of the base and square of the perpen- 
dicular together, and the square root of the sum will be the 
hypotenuse. 

Example. — The distance from the base of a building 
to the sill of an attic window — perpendicular of the 
triangle, B C — is 40 feet; what must be the length of 
a ladder, — hypotenuse of the triangle, A G, — placed 
on a level with the base of the building, and 12 feet 
therefrom, — base of the triangle, A B, — to reach to 
the sill of said window ? 

40'-^-i- 12- = V^l 744 =r 41 J feet. Ans, 

The hypotenuse and one of the sides of a right-angled triangle being 
given, to find the length of the other side. 

Rule. — Add the hypotenuse and the given leg together, mul- 
tiply the sum by the difference of their lengths, and the square root 
of the product will be the side sought. 

Example. — The sill of a window in a building standing on the 
edge of a stream is 30 feet above the water, and a line which has 
been extended therefrom directly across the stream to the opposite 
shore is found to measure 80 feet ; required the width of the stream 
at that place. 
80—30 = 50, and (80 -^ 30)X 50— v/5500 zz: 74.16 feet. Ans. 

To find the height of an inaccessible object, C, or the length of the 
perpendicular B C. 

Rule. — Upon a plane, at a right angle p 

to the base of the perpendicular, as A B, 
erect, at any convenient distance from the 
base, a perpendicular staff D E, and con- 
struct the hypotenuse A E in the direction 
A C ; then, as the base A D is to the per- 
pendicular staff D E, so is the base A B to 




the heio-ht B C. ^^ ^ ^ 



190 



MENSURATION OF SUPERFICES. 



Example. — The perpendicular D E, of the right-angled triangle 
A D E, being 3 feet, and the base, A D, 5 feet, what is the perpen- 
dicular of a similar triangle, A B (', whose base, A B, is 100 feet? 
5 : 3 :: 100 = 60 feet. Ans, 

To find tJie distance from a given point, B, to an inaccessible object ^ 

as cU A. 
Rule. — Draw a line B in the direction A, 
and frona the point B, at a right angle to B A, 
draw, to any convenient length, a line B C ; form 
a rif^ht angle to B C, from the point C, and con- 
vert into a triangle by carrying the hypotenuse, 
I) E, to the line 13 C, in the direction D A ; then, 
as the distance C E is to the distance C D, so is q 
the distance B E to the distance B A. 

Example. — The distance from C to E is 
rods, tbe distance from C to D, 15 rods, and the iJ 
distance from E to B, 22 rods, whereby required the distance from B 
to A. 

4 : 15 :: 22 = 831 rods. Ans, 




D gI 



OF QUADRILATERAL FIGURKS. 




n 11 





Tl 



Square. 



Rccian;;lc. 



D K 

Rhomboid. 



To find tlie area of either the above figures. 

Rule. — Multiply the length by the breadtli or perpondiculai 
height, and the product is the area. 

Example. — JIow many square feet in a floor whose form is a 
rhomb, or rhombus, each side being 12 feet, and the perpendicu- 
lar or right-angular distance from the side F G to the side H I being 
8 feet ? 

12X8 = 06 square feet. Ans. 



LINES AND SUPERFICIES. 191 

The above figures are parallelograms^ for a parallelogram is a 
quadrilateral whose opposite sides are equal. 

A square is an equilateral rectangular parallelogram : it is two 
equal and similar right-angled isosceles triangles by either of its 
diagonals ; draw both its diagonals, and it will be made up of four 
equal and similar right-angled isosceles triangles. 

A rectangle is a right-angled parallelogram that has more length 
than breadth : it is two equal and similar right-angled scalene tri- 
angles by either of its diagonals ; by both, it is two equal and 
similar obtuse-angled isosceles triangles, and two equal and similar 
acute-angled isosceles or equilateral triangles. 

A rhombus, rhomb, or lozenge is an equilateral oblique-angled 
parallelogram : it is two equal and similar obtuse-angled isosceles 
triangles by its longest diagonal ; it is two equal and similar acute- 
angled isosceles or equilateral triangles by its shortest diagonal; 
it is four equal and similar right-angled scalene triangles by both 
its diagonals. 

A rhomboid is an oblique-angled parallelogram that has more 
length than breadth : it is two equal and similar right-angled sca- 
lene triangles by either of its diagonals ; by both, it is two equal 
and similar obtuse-angled scalene triangles, and two equal and 
similar right-angled scalene triangles. 

To find the Area of a Trapezoid. 

Rule. — Multiply the sum of the two parallel sides by the per- 
pendicular distance between them, and divide the product by 2 ; 
the quotient will be the area. a b 

Example. — The side A B, of the trapezoid A B- 
C D, is 48^5^ feet, the side C D is 72^% feet, and the 
perpendicular distance between the sides is 40J feet ; 
how many feet area has the figure ? c 

72^ + 48^5^ =120fX40i = 48901-^2 = 2445 ft. 2^ in. Ans. 




D 



To find the Area of a Trapezium. 
Rule — Draw a diagonal through the figure, which will divide 
it into two triangles, and multiply the length of the diagonal by 
half the sum of the altitudes of the triangles ; the product will be 
the area. 

Example. — The diagonal A D, in the trapezium 
A B C D, is 54 rods in length, and the altitudes of 
the triangles formed by the introduction of the 
diagonal are 20 rods and 2G rods ; required the 
area of the figure. c 

26 + 20 -f- 2 = 23 X 54 = 1 242 square rods. Ans, 





192 MENSURATION OF SUPERFICIES.. 

Having thi figure of a rhotTibus, rhomboid^ or trapezoid, the perpen- 
dicular, ichercby to find the area, may be found by the following 
method. 

Suppose tlie diagram A B C D. 

A D and B C sides parallel to 
each other. 

A D base. 

A B side inclining to tlie Uase 
and whose length is known. 

Then, at any convenient dis- 
tancc from the angle A, on A I^^at x 
erect a pcrpendicuhir to A D that -"■ 
will cut the side A B, as i h ; th«in will A i be to i A as A B to B E, 
pcrpenrlicular retjuired ; or A i will be to A A as A B to A E, dis- 
tance from the angle A, on A D, at which a perpendicular dropped 
from B will fall ; and 

\/(A B' — A E") =BE, perpendicular required. 

Example. — The figure A B C D is that of a field whoso side A B 
is 08 rods : A i is 4.2 feet, and ih is 3.7 feet ; required the perpen- 
dicular distance from the side A D to the side B 0. 

4.2 : 3.7 : : G8 : 59.9 rods. Ans, 

ExAMru:. — A bin in the form of a trapezoid, A B C D (Fig.), has 
a side A B inclining to A D that is 12 feet in length, and a i>orpen- 
dicular to A I), erected on A D, that cuts the side A B, gives the 
Bognient A i 2 feet, and the segment A A 14 feet ; rccjuired the per- 
pendicular distance from the side A 1) to the side B C. 

2 : 1.5 : : 12 : 9, and V(12*^ — 0-) = 7.94 feet. Ans. 
A I : A B : : A A : A E, diagram. 
A E : AB : : AA : A I, 
A A : E A : : A t : B i, ** 
I A : A t : : B E : AB, 
A A : I A : : A E : B E, 

To find a diagonal of the above figure A B C D. 

V (BTr + Elr) = B D, diagonal. 

Having the ana of a rhombus^ rhomboid, or trapezoid, and th* .sum if 
any two sides of the figure that are parallel to each other, the pcrpen' 
dinilar to those sides icill be found, if we divide twice the area by the 
sum referred to. 

Suppose A 1) and B C tlie given sides ; 

B E the perpendicular required ; then 



MENSURATION OP SUPERFICIES. 193 

2 area area . _ ^ 

area 2 area _ 

, — AD==BC. -0-R- — BC=^AD. 



iBE~^^-^^- BE 

Tf through any four-sided figure a diagonal be drawn or he supposed 
to he drawn, the figure will he converted into two triangles, each of 
which will have a side, the diagonal, that will he common to them 
hoth ; and the length of that side or diagonal, which is an indispen^ 
sahle element in calculating the area of a trapezium, may he found, 
when more simple means may not he resorted to, hy one or the other 
of the following methods : — 

Suppose the figure represented by the diagram A B C D. 

B CI C to A, diagonal required. 

/s\ .'^^^/ Construct a partial diagonal, C t, 

\ ,,-'''' V; direction C A, and thereon erect a 

^\^<'' / perpendicular to C ^ that will cut 

^-''''' \ \ / an adjacent side, as r ^ cutting the 

^X\ / sideCD; thenCr :.C^: :CD:0 

— 1-:^ F, and C r : r ^ : : C D : D F ; and 

"- " /v/(AD^ — DF') =AF, andCF 

-j- A F = C A, diagonal sought. 

Example. — A structure in the form of a trapezium., A B C D, 
(Fig.) has C r 8 inches, C ^ 6 inches, and r t h inches ; the side 
C D is 16 feet, and the side A D is 20 feet ; required the length of 
a diagonal C A. 

8 : 6 : : 16 : 12 feet, C F, or distance from C, on a line C A, at 
which a perpendicular dropped from D will fall, and 

8 : 5 : : 16 : 10 feet, D F, or length of that perpendicular, then 

202 — 102 = V300 = 17.32 + 12 == 29.32 feet, length of a diag- 
onal C A, or of the side A C of the triangle A C B, or of the side 
C A of the triangle CAD. 

Again, suppose the same diagram, the side A D accessible, and 
D to B the diagonal required. 

Dn:DE::D5:DB, diagonal sought. 
17 



194 



MENSURATION OP 8UPERPICIM. 



OF POLYGONS. 
To find the area of a regular polygon. 

Rdlk. — Multiply tlie length of a side by half 
the distance from the side to the centre, and that 
product by the number of sides; the last product 
will be the area of the figure. 

p]x AMPLE. — The side A B of a regular hexagon 
is 12 inches, and the distance therefrom to the 
centre of the figure, d c, is 10 inches ; required 
the area of the hexagon. 

^ X 12 X 6 =« 3G0 sq. in. = 2j sq. feet. Arts. 

Table of angles relative to the construction of Regular Polygons with 
the aid of the Sector^ and of co-effiricnts to facilitate their construction 
without it ; also, of co-cfjiciaits to aid in finding the area of the figure 
the side only being given. 




S»mt: 


of 

•ides. 


Anfie 
cfnlr*. 


Ani^le 

at 

circum. 


beinf 1. 


L*ncth rf 
side, nidiut 
^ belncl. 


Radiua o( 

circle, tide 
Iring 1. 


Radiua of 


Arra, 

•idt 

be.ncl. 


Triangle, 
.Spiare, 
Pentagon, 
Hexagon, 


3 
4 
o 
6 


i2no 

72 
60 


60O 

90 
I OS 
120 


0.2S96:$ 
0.6 

0.6^ 
0.'566 


1 73205 
1.4142 
1.1765 


.r.773 
.7071 

.85(16 
1. 


2. 

1.4142 
1.236 
1.156 


0433013 

1. 

1.720477 
2 698076 


Heptagon, 


7 


51^ 


12-i 

135 
110 

144 


l.C0^2 


.8677 


1.152 


1.11 


1633812 


Of.laffon, 
Nonagon, 
Decaf on, 


H 
9 
10 


45 
40 
36 


1.2>>71 
1.3737 
1.63>S 


.7654 
.6>l 

.618 


1.3n65 
1.4G19 
1.6IS 


1.0823 

1.06 

1.05 


4.8S84V 
6.16I8M 
7.694900 


Undccagon. 


11 


32^ 


147^ 
150 


1.702S 


.66:m 


1.7747 


1.01 


9.36664 


Dodecagon, 


12 


30 


1.S66 


.6178 


1.9318 


1.038 


ll,196ia 



NoTB. — " Aneic at centre " means the angle of radii, passing from Ihe centre to tkt 

circmnference. or corners of the figure. 

" Angle al circtmifcrcnce " means Ihe angle which any two adjoining sidca make witk 
each other. 

Every circle contains its own radius, as a chord line, exactly six 
times ; therefore, 

To describe a polygon with the aid of the sector. 

Rule. — Take the chord of 60^ on the sector, and describe a circle; 
then, with the chord, (on the same line of the sector,) of as many 
degrees as indicated in the table, for the respective polygon — col- 
umn, ** angle at centre " — space off the circle, and each space will 
be the side of the polygon required. Thus, for a decagon^ take the 
chord of 60^ on the sector for the radius of the circle, and the chord 
of 30^ on the same line of the sectors for a side. 



MENStTKATION OF SUPEEHCES. 195 

Example. — 1. The radius of a circle is 7 feet ; required the side 
s®f the greatest regular octagon that may be inscribed therein. 
7 X .7653 = 5.3571 feet, or 5 ft. 4i in., nearly, Ans, 

2. The sides of a heptagon are to be, each, 5 inches ; required the 
radius of circumscribing circle. 

1.152 X 5 =^ 5.760 == 5| -j- in. Ans, 

3. Each side of a hexagon is 12 inches ; required the distance 
i^d c) from the centre of a side to the centre of the figure. 

12 X .866 = 10.392 inches. Ans, 

4. The side of an equilateral triangle is 12 inches^ required its 
altitude, — the perpendicular. 

12 X 3 X .28868 = 10,392. Ans, 

5. A Perpendicular, from the centre to either side of a hexagon, 
is required to be 12 inches ; what must be the radius of circumscribing 
circle ? 

12 X 1.156 = 13.872 inches. Ans, 

To find the area of a regular polygon, when the side only is given. 
Rule. — Multiply the square of the side by the number or factor 

in the table — (column, Area) — opposite the name of the respective 

polygon, and the product will be the area. 

Example. — Each side of a nonagon is 6J rods ; required its area. 
6.5"^ X 6.181824 = 261.18 + square rods. Ans. 

To find the <irea of an irregular polygon. 
Rule. — Divide the figure into trapeziums and trian- 
gles, by drawing diagonals, and find the area of each, 
separately ; the sum. of the several areas will equal the 
area of the^figure. 

Example. — The outline of the above figure defines an irregular 
polygon ; the enclosed lines divide it into three triangles ; and the 
areas of the several triangles, taken collectively, are equal. to, or con- 
stitute, the area sought. To find ike areas of the triangles, see Tri- 
angles — Meisuration of. 




196 LINES AND SUPERFICIES 



CIRCLE. 

A CIRCLE is an endless line equidistant ih all its parts from 3 
point within called the focus, or centre. 

It is generated when a plane revolves about a conic surface per- 
pendicular to the axis, or by a plane cutting through both sides of 
a cone parallel to the base. 

It is equated by means of the triangle, as we shall see. 

Eitlier semi-<liameter of two diameters drawn at right angles t 
(jach other in a circle may be an ordinate. 

In TuiGONOMKTRY, a circle is the mcrt^wrc of angles, and is sui 
posed to be divided into parts called degrees, minutes, and second 
also into semicircles, quadrant^, and arcs. In THIGONOM^:TI:^ 
arcs are the measures of angles, and angles are the measures of ar^ 
an^l sides (see Tkigonometky). 

In (iKoMKTUY, a circle-plane is usually called a circle, and its 
boundary line, above described, is called the circumference, or 
j)eripherv ; thus triangles, as they irlate to the circle, and are 
drawn within its circuud'crcnce, may be counted sectional lines of 
the circle, and treated as such. 

Generally, then, a circle is a plane figure having a circumfer- 
ence, a radius, a diameter, an area, &c. 

TIk; ratio of the circumference to the diameter of a circle is 
commonly expressed by the (ireek letter t (pi), and ^ will be used 
with that signiGcation and no other in this work. The exact value 
of TT, or the length of the circumference of a circle when the 
diameter is 1, it is well known has never been found: it can no 
more be written with nuujbers, probably, than can the exaet root 
of a surd number. 3.Mi:>:»2G5a.'i8:>793'i3Mr.2<M33H32795028841- 
97169391)37. , , =z tt. In practice, tt is usually taken = 3.1416. 

A circle contains a greater area than any other figure of the 
same length of perimeter or outline. 



MENSURATION OF SUPERFICIES. 



197 




OF THE CIRCLE AND ITS SECTIONS. 

Relationship of the sectional lines of the circle^ 

^ne with another, whereby, any two being 

given, any other may be found. 

To furnish several examples, and to exhibit 
the proof with the operation, suppose the lines 
a b and p d given, and let a 5 = 20, inches, 
feet,' yards — any linear measure, — and let 
pd=Z,b\. 
ah a p 

20 -7- 2 = 10, sine of half the arc, or half the chord of the segment, 

ap'^ p d^ p d 2c a 

100 + 12.32 -r- 3.51 = 32, diameter, 

2ca c a 

32 -^ 2 = 16, radius. 

c a p d cp 

16 — 3.51 = 12,49, cosine of half the are. 

ca c p p d 

16 — 12.49= 3.51, versed sine of half the arc, or height of segment, 

cp p d ca 
12.49 + 3.51 = 16, radius. 

ap"^ pd^ a d^ ad^ ad 

100+ 12.32 = 112.32, and V 112.32= 10,5981, eh'd of half the arc. 



ca^ 



ap^ 



cp"" 



cp'^ 



cp 



256 — 100 = 156, and V156 = 12.49, cosine, 

ca^ cp'^ ap^ ap'^ ap 

256 — 156 = 100, and VIOO = 10, half the chord of seg. 

cf- ap^ cd^ cd^ ca 

156 + 100 = 256, and /s/25Q = 16, radius, 

Radius and height of segment given, to find chord of segment, 
ca — p d=cp,dJ>Acd^ — c p^ =■ ^ a p^ ^=- ap, and apX 2 = 2 ap"^ 
chord, a b. Ans, 

Chord and height of segment given, to find diwneier. 
i a 6"-}- p d^= a d'\ and a d^ -^ p d =z 2 c a. Ans. 

Radius and height of segment given, to find chord of half the arc. 
c« — p d = cj9,and c a} — cp^ =aj9'^, and ap- ■\- p d^ =^^ a d^ =^ad 

17* 



Ans, 



198 LINES ANP SUFERnCIEg. 

Radius etnd sine given y to find versed sine, or h^igh/, 
td^ — ap'^^ cp^y ^nd c a — cp^^pd. Ans. 

Radius and cosijie given, to find chord of half the arc. 

ca^ — c p'^^= a p',c a — c p =^ p d, iinc} u p'^ -|- p d' s/lTd^ ^=taJ. 

Ans. 
Radius and chord of half the arc gian, lu jma sine of half thr ar'-. 
ad^~-2ca=pd, and a d^ — pd' =z /s/ap^ ^^ ap. A ns. 

Chord of half the arc and sine of half the arc given, to fmd radius 
• d^ — a;?^ = /? J"^, and a rf^ -f- 2/) </= cfl. Ans, 

Chord of lialf the arc and sine of half the arc given, to find cosine. 

ad^ — ap'^^= pd^, and ud^ -^ 2p d = c a, and c a* — 0"^ = 
^cp'^=cp. Ans. 

Radius and sine of half the arc gircn, to find nrscd sine. 
cd^ — a y = cp^, and ca — cp ^^pd. Ans. 

Sine ctnd Versed Sine given, to find Cosine. 

ap -j- ;? <r = n d" .ind a d^ -^ 2 p d = c a ; ca — p d= c p. Ans. 
The equation of the circle is commonly written y =a" — x. 
which, referred to the foregoing, is equivalent to the expressi* 

ap'^^ac — c/?", and denotes the same quantities. But, in An- 
alytical Geometry generally, sines and cosines are not known : thejr 
give place to nrdinates and abscissa:. Thus in the equation of the 
circle, and in constructing it by plotting the double ordinat«.*s of 
different abscissa, ap is an ordinate and cp is its abscissa. U 

ferring to the diagram, a ;> :=: 2 ac X p d — ;> a ♦ and this is an 
expression for the equation of the circle : it may be writti^n 
y=:\f(du — r^. Both r/> and dp are alwcissjc to the ordinate 
a p. The equation of a curve expresses the relation between the 
ordinate and corresponding abscissa of every point in the curves 

To find the Diameter, Circumference ^ and Area of a Circle. 

Let d represent the diameter, c the circumference, and •i tho 
area. 

J = C-^rrz=c-f-3.141G=v/(lA -^ r) — 1.12838v^A. 

c = TTd = d X 3.141G — 7rv(4A-^ ") = 3.uU0\^X 

A = 7r(/- -^ 4 =: (i" X .78j4 = cc/~4=:c^-^47r = ffr^. 



LINES AND StrPERFICIES. 199 

Example. — A lin-e (the chord a 6, diagram) was stretched 
across a circular railway 46 feet, and the versed sine, p d, was 
found to be 1.8 feet; required the diarnqter, circumference, and 
area of the circle. 

46 -^ 2 =1 23, and (23^ -f 1.8^) -f- 1.8 = 295.6$ feet, diameter. 

Ans. 
. 295.69 X 3.1416 =928.94 feet, circumference. Ans, 

295.69^ X 0.7854, or ^ 68669.55 square feet, or 

928.94 y 295.69 | 68669.55-^-43560 = 1.576 acres ^' 

To find the Length of an Arc of a Circle, 

Let r represent the radius of the arc ; n, the number of degrees 
in the arc ; J, the diameter of the circle to which the arei)elongs; 
t\ the versed sine of half the arc ; and Z, the length of the arc. 

/— -/ . t7 , Sy^ , 3.5.v^ , 3.5.7.??^ , \ 

\ ^2.3.^ 2.4.5.C?* 2A.e.7.d^ 2.4.6.8.9.d* / 

I = nrn -^ 180 z= .01 745329252rn. 

These rules are applicable to allarcs of circles, whether greater 
or less than half the circumference. But the former is too te- 
dious for general practice, and it is not often that we can know 
the number of degees in the arc without the aid of trigonometri- 
cal tables. We can approximate the length with certainty and 
considerable precision by other methods. 

Let b represent the chord of the arc, a b, diagram ; ?-, the radius 
of the arc ; c, the cosine of half the arc of the salient angle, cp, 
diagram ; Z, the length of the arc of the salient angle ; and 

jt |:=(l+iV^= 86.13961: then 

When c is equal to or greater than ^b, 
Z = 1.50341985r-f-(r + ^c+ ^=^) very nearly, 
and 7i=zkb-^ fr -\- ^c -}- ^~| j very nearly ; 
butZ = ^; .•. = J^^_^|i^^yery nearly. 



200 LINE8 ilXD BnPERFlCil3, 

When c is less than -Ji, 

/ O.D57107(! \ , 

Z = 7177 1 n — I ^cry nearly^ 

V 'r + iO + i!^) 

and n = 180— ^^^ vcrv noarlv ; 

- + i^ + ^^ 



./= 



^(^''-TT^nj)^-^'^"^"'^- 



But 2;rr 7= clrcumfcirncc of circle containing 360°. 
The length of the greater arc, or arc of the re-entrant angle, 
therefore z=z ^-r — '. nnd contains 3G0 — n degrees, . • . 

^ «r (360 — ii) -T- 180. 

Now, by the foregoing expressions for n, we obtain the calco* 

lated angles kh ^(j'-\-^c + ^^') , and 2kc -^ (r + } 6 + ^^) , 

strictly correct for angles 45°, 90°, 135°^ an<l 180°; and the errors 
are e(|ual and rccipro<*al for angles and their supplcnK'nts of 180°. 
They are all — from to 45°, and from 1)0° to 135°, and -|- from 
45° to 00°, antl from 135° to 180°. The maximum error is only 
2'. It ol)tains when c and ^b are to each other as 3 to 4, or at 
78° 44' 22'' antl H)G° 15' 38". The mean error (one minute) oc- 
curs at 16° 15' 38", 28° 44' 22", 57° 28' 44", 86° 13' 6", 93° 46' 
54", 122° 31' 16", 151° 15° 38", and 163^ 44' 22". 

NoTK. — The popular cxpresgloii for tlic Icn/irtli of the arc of a circle, when* 
the ratio of the arc to tin- rircumfcrcnco is unknown: viz. 'Srn — ^)-+-3, In 
"which m ropn-smts the choni of half tlio arc, and b the r) . arc, an- 

ewers tolerably well when thr arc is not greater than a., .r it ex- 

presses a (iiiadrant, only V 'JO" sliort, and for a shorUr ai- .i r i> h-ssi 

out it expresses the areas' short when it is equal to two qutulrauU. 

Example. — The chord of a circle-arc is 24 and the cosine 16. 
What is the lengtii of the arc ? 

r = V[c-^ + (ihf2 = 20, and c > ^b ; 
*!, 86.13961 X 24 

then « = 20 + r+70?32l8-l= '3°-^'27<8, 

, ,— ^rn -,..,„., 1.50341S3X 21X20 
and l-.^- =25.-olG2=-^3-^p^-^ ~2j-j , 

and this example involves the maximum oiTor, 
(true /= 25.7399vS). A us. 



LINES AND SUPERFICIES, 201 

Example. — Half the chord of a circle-arc is 9, and the versed 
sine 5 ; required the length of the arc. 

(9^ -f- 5^) -^ 2 X 5 = 10.6 = radius of arc, 
and 10.6— 5 = 5.6 = cosine, c<^j6; 

then fi = 180 172.27922 X 5.6 — ^u6°.191765, 

10.6 +4.5 + .0197354 

^7 '^rn „, ,^^ A .957107X5.6 \ 

To find the Area of a Sector of a Circle. 

Let A represent area of sector of salient angle (achd a^ dia- 
gram) ; Z, length of its arc ', n, number of degrees in its arc ; r, radius 
of circle. 

A — Trr^i + 360 = ^^, 

But ^r^zzzarea of circle containing 360°, 

The area of the greater sector, or sector of 

the re-entrant angle, therefore = 

9rr-(360 — n) + 360 = Trr^— i/r =z Trr^— A, 

To find ike Area of a Segment of a Circle, 

Let A represent area of segment of salient angle (ah d a^ dia- 
gram) ^ Z, length of its arc ; re, number of degrees in its arc ; c, cosine 
of half its arc (c p, diagram) ; &, chord of arc ; and r, radius : then 

A = ^r;i -^ 360— i5c = ^Ir^^hc, 

But the area of a circle zn ^i^. 

The area of the segment of the re-entrant angle therefore zz: 

:rr-(360 — ?i) + 360 + ^hc = ^rr- + ibc—^Ir = 7<r_ A. 

Note. — When the arc of the segment does not contain more than 90*, 

A = — (-^4-6) nearly, = -!^ : nearly ; m being the chord of half the 

10 \ 3 y Ob 

arc, and v the versed sine. And when the arc contains more than 90°, and not 
exceeding 180,° A=~ 's/lp^ ^ — ) nearly. 




- 1 h d beiDg tbe diameter of the circle, 



202 LINES AND SUPERHCIES, 

To find the Area of a Zone. 
Rule. — Find the area of the circle containing the zone, and 
the areas of the segments on the sides of the zone, and from tbe 
area of the circle subtract the sum of the areas of the segments ; 
the (lifrerence will be the area sought. The rule is also applicable 
in finding the area of a zone of an ellipse. 

To find the Diameter of a Circle of which a given Zone is a part. 

B the greater base, and b the less, of the zone, and h the perpen- 
dicular distance between the bases. 

To find the Area of a Crescent. 

RiTLF. — Find the areas of the two scgmcntjj formed by the 
arcs of the crescent and tluir chord ; subtract the less from the 
greater, and the dilTerence will bi^. the area of the cresci't 

To find the Side of a Square thai shall contain an And tfjum to 
that of a given Circle. 

Rule. — Multiply the- diameter of the circle by \/(t-1-4) = 

.886228, or multiply the circumference by \/:r-i- 2t z=l .28209447; 
the product will be the side. 

To find the Diameter of a Circle that shall have an Area equal 
that of a given Square. 

Rule. — Multiply the side of tbe square by \/(4-^7r)zz: 
1.128378, and tbe product will be the diameter rer|uired. 

To find the Diameters of Three Equal Circles the greatest that can 
be inscribed in a given Circle. 

Rule. — ^Multiply the diameter of tbe circumscribing circle ly 
3 -^ ^T' zz: .45594322 ; the product will be tbe diameters requir* 

To find the Diameters of Four Equal Circles the greatest that can 
be inscribed in a given Circle. 

Rule. — Multiply the diameter of the given circle by 1 -f- rr^^i 
.4052829, and tbe product will be the diameters sought. 



LINES AND SUPERFICIES. 203 

To find the Side of a Square inscribed in a given Circle. 

A square is said to be inscribed in a circle when all its corners 
touch the circumference. Its area is half that of a circumscribing 
square of the same circle, 

KuLE. — Multiply the diameter of the circle by ^^ = ^ a/2iiz 

.70710^8, or multiply the circumference by y'^-^7ri=.225079 ; 

the product will be the side of the inscribed square. 

Example. — A log is 28 inches in diameter at the smaller end ; 

required the side of the greatest square that can be hewn from it. 

28 X .7071068 = 19.799 inches. Ans. 

Note. — "With reference to circumscribed and inscribed figures generally, 
see Centres of Surfaces, p. 291. 

To find the Diameter of a Circle that will circumscribe a given Tri- 
angle. 
Let I =z longest side of triangle ; m, intermediate side , s, shortest 
side ; d, diameter of circle. 

ms 

When two of the sides are equal, 
2a 

a being one of the equal sides, and ?« the unequal side. 
When all the sides are equal, 

d=z 5T = = 1.1547a. 

V(3aV \/S 

To find theDiameter of the greatest Circle thai can be inscribed in 

a given Triangle. 

I -\- m -\- s 
When two of the sides are equal, 

J m \/ (4a^ — m^) 

7n -)- 2a 
When all the sides are equal, 

dz::z\/(Sa^)-^3 = a-^\/ 3 = .577- 



204 



IIHS9 AXD StrPERFICI£». 



To divide a Circle into any number of Concentric Circles o/eqmatmrttf. 
Rule. — 1. Multiply the square of the radius of the piven circle 
by the numlxir of conccDtrio circles, less 1, required, ani divide the 
product by the nuinlx'r of concentric circles rc(juired; the sc^uare 
root of the quotient will be the radius of the given circle, less tlit 
breadth of the outennost concentric circle. 

2. Multiply the yjuare of the radius of the piven cirde by the 
number of concentric circles^ lej^s 2, requireil, and divide the product 
by the number of concentric circles recpiircd ; the wjuare root of 
the (juotient will l)e the radius of the given circle, less the sum of 
the breadths of the two outermost concentric circles, &c. 

ExAMi'LK. — Ili'fjuired to divide a circle of 20 inches raifii: 
into two concentric circles of equal areas. 

20^-^ 2 = y/ 200 = 14.142 inchei, radios of inner circle. 
20 — 14.142 = 5.h:)H inches, breadth of outer cin-lc. 
The followiiirr diagram illustrates the principle of the foregoin 
nde, and cxliibits tne mrchanical mctho<l of solving the pnM)leii, 
It is perhaps unnecessary to add that it is wholly immaterial in! 
how many concentric circles of e(}ual areas the given cindc is to be 
dividi*d, or whether the concentric circles an* to have e<jual areas 
or otlicrwise. The radius of the rrivcn circle has oidv to fk' divided 
into the requireil number of a!i(]uot parts, or nrojKirtional |K'irts,thc 
parallel lines struck, the curves drawn, and tiie division is accom- 
plished. 

Example. — Four men, A, B, C, and 1>, own a grindstone, r> 
15 inches ra^lius, equally between them. A is first to grind (»:i 

his share, then B, then C 
and 1> Ls to have the r« 
n»ainfk»r. Uecpiired tli 
number of inches tliat nia 
be CTOund frora the radii i 
by A, B, and C respect ivi 
ly, and the ra<lius of th« 
stone that will l>e left for ]> 
V/(15»X 8 — 4) = 12.90. 
and 15 — 12.9D = 2.01 in 
—A. 

y/(15-X2^4)= 1" 
and 12.90— 1 O.GOG = _ 
inches, — B. 
^(lo=-i-4)iz:7.5, and 
10.606 — 7.5 = 3.106 in.,— C. 

15 — (2.01 + 2.384 -f 3.106) = l.b inches,— D. 

NoTK. — Tlie square root of the quotient obtained by dividing the square of 
the railius of tlie given circle by the number of concentric ciracs required Is 
equal to the radius of the innermost concentric circle. 




LINES AND SUPERFICIES. 



205 



To find the area of the space contained between two Concentric 

Circles. 
Rule. — Multiply the sura of the two di- 
ametei^ (that of the outer circle and 
that of the inner) by their difference, 
multiplied by .7854, and the product will be 
the area of the ring, or space, referred to. 

Example. — The diameter of the larger 
circle, A B, is 36 inches, and the diameter 
of the less circle, C D, is 30 inches ; required 
the area of the ring, or space, between the 
two circles, 

36 + 30=z66, and 36 — 30=6; then 
Q(jX Q X .785411=311 square inches. Ans. 




ELLIPSE. 

An ELLIPSE is a single closed curve and an isometrical projec- 
tion of a circle. 

It is formed when a plane cuts through both sides of a cone or a 
cylinder obliquely to the base. 

It has two foci : they are on the transverse diameter, one on 
each side of the centre of the figure ; and they are equally distant 
from that point. 

An ellipse is divisible by either of its right diameters into equal 
and similar semi-ellipses, and by both into four equal and similar 
elliptic curves. 

The extremities of the two diameters and two parameters of an 
ellipse are eight points in the. circumference. The summit of every 
ordinate is a point in the curve. The summits of every double 
ordinate are two points in the circumference. 

Either semi-diameter of an ellipse may be an ordinate, though 
the axis of abscissae proper is the major axis. 



Let ^ 1= axis of abscissas. 
d == axis of ordinates. 
V = greater abscissa. 
X := lesser abscissa. 
z = abscissa from centre. 
y = ordinate. 

18 



Let t zn transverse diameter. 

ciz: conjugate diameter. 

p z= parameter. 

fzn. distance from vertex to focus. 

g zz: space between the foci. 

e zzL eccentricity. 



206 



LINS8 AND S0P£EFICII». 



When the transverse axis is the axis of abscissas, 
f-L,(l^^=vxz=itv — v^z=itX'-'2^ = \i*'-z*; also 

When the conjugate axis is the axis of abscisses, 

2^«(1 _ fcS) — vx z=. cv — r' z= ex — z' = i c* — 2* ; also 

/,— ^ — w — ^ = *y 

r=|+Jv'(i9'-jO=^''.^' = J« + = = ^-*, 

«=:Jv/(iC*'-y») = 4^-i=i-J^ = 4.;— Jx, 

y=%(rx) = ? (v^^r - r^) = ^Y(^x- X-) =ia\^f'-^"' 
^ ^ 9 ♦ 

Example. — TIic cccentririty of an ollipv is 0.8, an onllnatc 
is 12, and ite abscissa, rcckoiKMl from tho centre, is 9.6; required 
the axis o^ al>scissa, and axis of ordinate. 

2v^(12' X (1 — .8-) 4- 9.0*) r= -:' J . axl^ of absciss*. Ans, 
|12'X(1— .8') + 9.6' 



>!' 



K1-.8') 

Example. — In an ellipse, a c h d^ the 
transverse axis, a h, is 40, the conjutrate axis, 
cd, 24, the ^eatcr abscissa, a p, 26, and the 
lesser abscissa, bp, 14 ; required the length 
of the ordinate ep. 



= lU, axis of ordinate. Am. 




LINES AND SUPERFICIES. 207 

£2=1 — (24^ -f. 40') = 0.64, and 
V/[(l — .64)X26X14], or ^ 

242 X 26 X 14 24v/(26 X 14) V = 11.4473. Ans. 



J 



40*^ 40 ' ) 

To find the Area of an Ellipse. 
A = 7ric-7-4 = 0.7854 tc. 



Example. — The axes of an ellipse are 40 feet and 20 feet. 
What is the area ? 

40 X 20 X .7854 = 628.32 square feet. Ans. 
l = itt(l 



To find the Length of the Circumference of an Ellipse. 



2^ 2-. 4' 21 41 6- 2''. 4-. 6-. 8' 

1\ 31 51 72. 9. e^" 



&n. 

2-. 41 6-^. 8-. 10^ 

But the following empirical formula, which I have been at some 
pains to devise, expresses the perimeter of an ellipse sufficiently 
near for practical purposes in all ordinary cases of eccentricity ; 
viz. — 



'=-j(?+^> 



And it may be proper to state that the expression furnishes I too 
short by 1-238 for ellipses of greatest and least possible eccen- 
tricities ; too short by 1-351 when c=^t; affords almost strict 
accuracy when c is not greater than |^, nor less than ^t ; and 
gives I too long by 1-274 when c = \t. It may also be proper 
to state that the empirical formula of the hand-books; viz., 

/ = TT —- — -, and which, by the way, I have seldom met with 






given other than as affording strict accuracy, affords very incorrect 
results for ellipses of much eccentricity. It is correct only for the 
circle, or when t and c are equal ; and therefore gives I in excess 
in all cases. 

To find the Area of an Elliptic Segment. 

The area of an elliptic segment is to that of a corresponding 
circular segment as the axis of ordinate is to the axis of abscissa, 
nearly. 



208 LINES AND SUPERFICISfl. 

Let = axis of abscissa. 

d = axis of ordinate. 

z = abscissa from centre. 

X = abscissa from vertex. 
B =^ base of corresp6nding circular segment. 
L =len;zth of arc of lesser corresponding circular segment 

A =^ area of lesser elliptic segment. 

/Ji=zv/(9^_4.«)=2v^(fc-x');z = i^-a% 

When z is equal to or greater than JB, 

_ 1. 5023^5 , 3.0046^/? , 

Lzi^ — , ver}' ncarlv, ==z ., - very nearly. 

-f- 2 . \\<^ — 2j ' ' 

When z is less than ^IJ, 

very nearly. 
. = l(i,L-iZJ.-) =?L^^-^^il^); and 

— A = area of greater elliptic segment. 

KxAMPu:. — The li.ise h of an elliptic se^rment is 32; the h< ' 
5; and the axis parallel to the base, 40. What is the area ol 
segment ? 

^ = l"_Xi 25- "^ 



2 z= (25 — 2 X "i) ~ 2 = 7.5 = J9 — X, 

B = v/(25- — 1 X 7.r,) = 20 = yj(<^ — 4r), 

Z=: ^-5 708 X 25 + .7854 X 20 — 3.0046 X 7.5 ^ 03 ^y^ 



1.4 



^ ^ 40(25 X 23.174 ~ 2 X 20 X 7.5) _ 

4X25 —111.**. ^T«. 

Or, in this case (see note p. 201), 



LINES AND SUPERFICIES. 209 



PARABOLA. 

A Parabola is a single open curve of two branches. It has 
but one focus ; and its eccentricity is constant, and equal to unity, 
or 1. 

It is formed, when a plane intersects a conic surface, in the di- 
rection of the base, parallel to an element of the surface, or par- 
allel to one of the sides. 

It is equated upon the principle of opposite parabolas having 
their vertices towards one another, and has both a transverse and 
a conjugate diameter ; but these lie outside the figure. The trans- 
verse is the distance between the origins of the opposite parabolas ; 
and the conjugate is tangent to the vertex of the true or contem- 
plated parabola, and is bisected by the transverse. The middle 
of the transverse is the centre, and its projection is the axis of the 
figure. 

But, in the parabola, different abscissae on its axis, reckoned 
from the origin, are to oi^e another as the squares of their ordi- 
nates. The parabola, therefore, may be equated independent of 
the supposed diameters and abscissae from the centre. Formulas 
will be given covering both methods. 

In geometry, any double ordinate of a parabola is the base of 
a parabola, and its abscissa, reckoned from the origin, is the axis or 
height ; thus a parabola is a plane figure having an area, a length 
of curve, &c. 



Let t = transverse diameter. 
c = conjugate diameter. 
p =z parameter. 
/;zz distance from origin to 
focus. 



Let X = abscissa from origin. 
z = abscissa from centre. 
y zzz ordinate. 



2 2 2 o 2 2 

c c cx 2y c z 



< = -=T-.= -^=22 



18* 



210 



LINK3 AND SUPERFICIBS. 






^ 






P 

2 






, iP(2s — 



To find the Area of a Parabola. 
A = \yx -^ 3 = 2W* -^ 3 ; h being the base and h the height 

7\> find the Area of a Zone of a Pnrahola, 



^; bi'inir (lie .uriator l»aso, // the lesser, and ^^ 

fz the (li:?taiicc between tlieiii. / h 



rr\ 



To find the Altitude of a Parabola, b, d, and a being given. 

k = l/a-^{}r — d^)' 

To find the Lcnfjth of the Curve of a Scmi-parahola, or tin T - fnth 
of a Semi-parabola. 

/ = V^(a:- + // " -f \xf/), nearly. 

NoTF.. — The common oxi)rcssion for the approximate lenpth of a '^ - i 

{parabola, viz. VCU*- -f-//-\ consitlirably exceeds the true len^rth, unle— , 
arge, compared with .r. When x is to ?/ as 3 to 5, it agrees with the foregmuf,' ; 
and, when y is not less than 4 times greater than x, it is entitled to the prefer- 
ence. 



LINES AND SUPERFICIES. 211 



HYPERBOLA. 



A Hyperbola, like a parabola, Is a single open curve of two 
branches, and has but one focus ; but its eccentricity is greater than 
unity, and hyperbolas may have different eccentricities. 

It is formed, when a plane intersects a conic surface in the direc- 
tion of the base, at any angle to the base between that which 
would generate a triangle and that which would generate a para- 
bola; or, in other words, at any angle short of 90^ to the base, and 
greater than that of the side to the base, — at any angle to the 
axis less than that of the side and axis. 

It is equated upon the principle of opposite hyperbolas having 
their vertices towards one another, as explained for the parabola. 

In geometry, any double ordinate of a hyperbola is the base of 
a hyperbola, and the prolongation of the transverse diameter is the 
axis^ or altitude ; thus the hyperbola is a plane figure, having an 
area, a length of curve, &c. 



Let ^ = transverse diameter. 
c= conjugate diameter. 
p = parameter. 
y= distance from origin to 
focus. 



Let X = abscissa from origin. 
2: =1: abscissa from centre. 
y z=: ordinate. 
e=z eccentricity. 






^~ 2 2 2 



■ ~" 2 * \ ' £- — 1/ 2 2c 2 2p 



_2c 
P 
tQp -I- 4^0 



-¥' 



Ap 



212 



LINES AND SUPE11FICIE8. 



y^{r^\){tx + x^) = px+x{e'-\) = 



C)/0^ + ^ 



\ t 2t ' 

To find the Length of a Semi-hyperholcu 



I = y (x- + IT -\r \xy) nearly. 
To find the Area of a IlyperhoUu 

A =: ~ \j^r "I" ^0 nearly ; h beincr the 

altitude, h the base, and m the ordinate at 
one-third the altitude from the vertex. 



Note. — The common expression for the Approximate length of a h>'perbolaAl 



viz., 22/ X ,T* 1 ,T/'T«»</' affords very incorrect reiolts when x if grew 
compared with 2/; but the empirical equation in tho bookB, vii.: — 




fbr tho hyperbola, agrees very nearly with that offered abore. 



I? 



CYCLOID. 

A Cycloid is an elliptic arch, whose span is equal to the cir- 
cumfcrcnce of the peneratlnrr circle, and rise enual to the diameter 
of tliat circle ; or, in other wonls, it is a scmi-cllipse by the trans- 
verse axis, when the axes are to each other as tt to 2. 

It is nrencratcd by a point in the circumference of a circle in the 
point of rest on a plane, when the circle makes one revolution 
from that point in a straight line. 

Its eccentricity is constant, the square of which is expressed by 

1--1- =.594717 + 

Its foci are in the span of the ardi, one on each side of the cen- 
tre, and they arc equally distant from that point : they are distant 
from their respective extremities of the curve, equal to 

2 "" 2 



LINES AND SUPERFICIES. 213 

Its parameter is a single ordinate, and is expressed by _ . 
Its equation is expressed hjy=. '^^x^ird — xj ^jj-^|^ 



ao^rees 



c I ' 

with y =z—\x{t — x) for the ellipse, when the diameters are, one 

to the other, as tt to 2. 

An Epicycloid is a curve generated by a point in the circum- 
ference of a circle, which revolves about another circle, either on 
the convexity or concavity of its circumference. 
- The cycloid, geometrically considered, has a perimeter, an area, 
and a length of curve. 

Let cZ, as before, equal the diameter of the generating circle ; 
then Z = 4.081J, and A == ttVP -~ 4=: 2.4674125^1 

Note. — I have been thus particular in treating of the cycloid, because I 
differ in my view from some writers with regard to the law of the curve. It 
is generally stated by writers on the subject, and without further specifyinff 
the nature of the curve, that its length is equal to 4 times the diameter of the 
generating circle, and that the area is equal to 3 times the area of that circle : 
and they do not compute either by the rules they furnish as applicable to the 
ellipse. If the cycloid is not strictly elliptical, it is clear, without comment, 
that I am ignorant of its law, and that the foregoing is inapplicable. But to 
prove that the curve is not strictly elliptical, -sfould be to prove — would it 
not ?-— that it is not generated by a point in the circumference of a circle 
rotating in a straight line on a plane. 

To find tJie Distance of Objects at Sea, Sfc. 

The CURVATURE of the earth, at its mean radius of 3956 stat- 
ute miles, or at 45^ of latitude, is [v/(3956- -f 1^) — 3956] X 
5280 = 0.66734075 feet (8.0081 inches) in a single statute mile on 
the tangent, and is as the square of the distance or space between 
two levels. For a geographical mile on the tangent, therefore, it 
is (6086 -f- 5280)- X 0.66734075 == 0.886043 feet. The mean hor- 
izontal refraction on the water is about ^^ of the curvature for 
any given distance ; thus the practical or apparent curvature is 
(1 — tV)' X .66734075 = 0.57541 feet for a single statute mile on 
the tangent. 

Prob. 1.-— What is the tangent from the place of observation to 
the sea-horizon, the elevation being 30 feet ? 

V/(30 -^ .5754) = 7.22 miles. Ans. . 
Prob. 2. — The distance from New York to Sandy Hook is 18 
miles ; at what elevation above the surface of the water, at either 
place, may the surface be seen at the other ? 

18^ X 0.5754 = 186.43 feet. Ans. 
Prob. 3. —'From an eminence 180 feet above the surface of the 



214 LINES AND SUPERFICIKS. 

water at the place of observation, the flag of a ship is seen in thd 
line of the sea-horizon, and the flag is known to be 60 feet above 
the surface where it is situati'd ; required the distance (tangent) 
from the observer's eye to the Hag. 




GO 

= 24.9 

^.5754 



miles. Ans. 



PuOB. 4. — From an efevation 60 feet above the surface of tli 
water at tlie place of observation, a vessel is seen in full view, and 
a portion of her canvas, .supposed to be 20 feet alwve the surface 
where she lies, is seen in the line of the sea-horizon ; what U the 
distance from the |X)int of observation to the vessel ? 

J CO I 20 ^^ ., 



STEREOMETRY, OR MENSURATION OF SOLIDS. 



OF PARAI.LELOPIPEDS AND CUBES. 
To find the lateral surface of a prism. 
HiLK. — Multiply the perimeter of the base by the 
allitu(U% and thr product is the lutoral surface. If the 
/surface of the riuire fij^ure is required, add the areas of 
the ends to the lateral surface. 

ExA.MPLE. — The sides of a Irianjjular prism are each 
2.t feet wide, and the length of cither side is 10 feet; 
required its hiteral surface. 

'-'i X 3 = Oi = periinotcr of base, and O] X IG = 108 
square feet. Ans. 

Example. — A hexajjonal prism has an altitude of 
12 feet, two of its sides are 2 feet wide each, three 
arc l-i foot wide each, and the remaining side is9 inches; 
required the lateral surface of the prism. 
2X2 = 4 
1.5 X 3 =^1.5 
.75X1= .75 = 9.'J5 X 1-' = Hi sq. feet. Ans. 

To find the solidity of a prism. 

Rule. — Multiply the area of the base by the altitude, and the 

product is the solidity. 

Example. — The length of a triangular prism is 12 feet, and each 
side of its base is 2i feet ; required its solidity. 



MENSURATION OF SOLIDS. 



215 



^ 2^5 _|_ 2.5 X -^■- = 2. 165-}- = sine of angle, or depth of base ; or, 

2.5 X 3 X .28868* = 2.165+ = height of triangle ; and 

2.165 X %-^ = 2.706-f = area of base ; or, 

2.52 X .433012* = 2.706325 = area of base ; and 

2.706+ X 12 = 32.475+ cubic feet. ^715. 

To find the solidity of a right prism or cube. 
Rule. — Cube one of its edges. 

Example. — The length of a side of a right prism is 
16 feet ; required its solidity. 

16 X 16 X 16 = 4096 cubic feet. Ans. 

To find the solidity of a parallelopipedon. 
Rule. — Multiply the length by the breadth, and 
that product by the height ; the last product will 
be the solidity. 

Example. — A slab of marble is 8 feet long, 3 
feet wide, and 6 inches thick ; required its cubic 
contents. 8 X 3 X -5 = 12 feet. Ans. 





OF PYRAMIDS. 
To find the lateral surface of a regular pyramid. 
Rule. — Multiply the perimeter of the base by half the slant 
height of the figure — half the length of a side — and the product is 
the surface of the sides. The surface of the entire figure is the sur- 
face of the sides with the area of the base added thereto. 

Example. — A triangular pyramid has a slant height of 60 feet, 
and each edge of its base is 20 feet ; required its lateral surface. 
20 X 3 X %^- = 1800 square feet. Ans. 



To find the solidity of a pyramid. 
Rule. — Multiply the area of the base by \ the per- 
pendicular height, and the product is the solidity. 

Example. — A quadrifateral pyramid has a perpen- 
dicular height of 21 feet, and each side of its base is 8 
feet ; what are its cubic dimensions ? 

8'^ X ^J- = 448 feet. Ans. 



* See Table of co-efficienls, etc., relative to Polygons. 




216 MENSURATION OF SOLIDS. 

To find the lateral surface of a Frustum of a Pyramid, Frustum of 
a Cone, Prismoid, or Wedgt. 

Rule. — Multiply the sum of the perimeters ot the bases by half 
the slant height. 

Example. — Each edjze of one of the bases of the frustum of a 
hexa^^onal pyramid is 4 feet ; each ed^re of the other base, 2 feet ; 
and the slant height of the frustum is 24 feet : required its bteral 

surface. 




4X0 + 2X^X ^2" = 432 square feet. Ans. 

To find the solidity of the fnistum of a pyrconid. 

Rule. — To the square root of the j)n)duct of tlie areas 
of the bases, add the areas of the ha.«<es, and multiply 
the sum by i the pcr|)endicular height of the frutJlum ;• 
the product will be the solidity. 

ExAMPLK. — The ^Tcater base of the fnistum of a 
quadrilateral pyramid is 3 feet on each side, and the less 
base is 2 feel on each side, the jH^rpendicular hcif^ht of 
the frustum is 15 feet; what are its solid contents? 

3X3 = 9 feet = area of preatcr base. 

2X2 = 4 feet = area of less base. 

15 -^ 3 = 5 = J height. Then, 

\) X 4 = /v/30 = 6 + 9 + 4«»19X5 = 95 feet. An*. 

OF PRISMOIDS AND THE WEDGE. 
To find the solidity of a prismoid. 

The RrLK for finding the solidity of the frustum of a pyramid is 
equally applicable to the prismoid. Or, 

Kile. — Add the areius of the ends, and four times the area of the 
mean between the ends, together, and multiply the sum by ^ the per- 
pendicular height ; the product will be the capacity or solidity. 

liXAMPLE. — The bottom of a rectangidar cistern is 8 feet by 
feet, the top is 4 by 3 feet, and the perpendicular depth is 12 feet ; 
required its capacity. 

8.6 

4 .3 

I'J . !) — 2 = 6 X 4.5 = 27 X 1 = 10."^ = 4 area of mean ; and 
6 X 8 + 4 X 3 = 60 + 108 = 168 X J^- = 336 cubic feel. Ans. 

* For aaother rule, sco Mensuralion of PriarooUli. 



MENSURATION OF SOLIDS 



217 



To find the solidity of a wedge. 
Rule. — Multiply the sum of the length of 
the edge and twice the length of the base, by 
the breadth of the base multiplied by the per- 
pendicular depth or length of the wedge, and 
divide the product by 6 ; the quotient will be the 
solidity. 

Example. — The length of the base of a 
wedge, a c, is 10 inches, the length of the edge, 
ef, is 8 inches, the breadth of the base, a b, is 
4 inches, and the perpendicular depth, d e, is 12 
inches ; required the contents. 

8 + 10 X 2 = 28 X 4 X 12 = 1344 -f- 6 = 224 cubic in. Ans, 




OF CYLINDEFvS. 

To find the convex surface of a cylinder. 
Rule. — Multiply the circumference by the length, 
and the product will be the convex surface. If the sur- 
face of the entire cylinder is required, add the areas of 
the ends to the convex surface. 

To find the solidity of a cylinder. 

Rule. — Multiply the area of an end by the length of 
the cylinder, and the product is the solidity. 

Example. — The diameter of a cylinder is 6 feet, 
and its lenfrth is 8 feet ; required its solidity or capac- 
ity. 

6 X 6 = 36 X ."854 = 28.2744 = area of end, and 
28.2744 X 8 = 226_2^— cubic feet. Ans. 

To find the length of a helix, or spiral, wound round a cylinder. 
Rule. — IMultiply the circumference of the cylinder by the number 
of revolutions the spiral makes around it, square the product, and 
thereto add the square of the length of the cylinder ; the square root 
of the sum is the length of the spiral. The rule is applicable in 
finding the length of the thread of a screw, hand-rail to a winding 
staircase, &c., &c. 

19 




218 



MENSURATION OF SOLIDS. 



OF CONES. 

To find the convex surface of a cone. 

Rule. — Multiply the circumference of the base by half the slant 

height, and the product is the surface required. If the surface of the 

entire figure is required, add the area of the base to the convex sui* 

face. 

Example. — The diameter of the base of a right cone 
is 8 feet, and the slant height is 18 feet ; required the 
convex surface. 

8 X 3.1ilG = 25.1328 = circumference of base, and 
25.1328 X ^- = 220-2^ sq. feet. Ans. 



To find the solidity of a con*.. 

Rule. — Multiply the area of the base by i the jjer- 
pendicular hci^'ht, and the product is the solidity. 

Example. — The diameter of the base of a right cone 
is 8 feet, and the perpendicular height of the cone is 15 
feet ; recjuircd its solidity. 

8'^ X .'^B5l X J»»^ «• 251J cubic feet. Ans. 




To find the solidity or capacity of the frustum of a cone. 

Rule. — 1. To the square root of the product of the areas of iho 
ends add tho iiroas of the ends, and multiply the sum by \ the perpen- 
dicuhir height (»f the frustum ; the product will be the solidity. 

Or, — 2. Divide the dilTerence of the cubes of the diameters of 
the ends by the dilDTonce of the diameters of the ends, and multiply 
the quotient by ) the perpendicular height of the frustum; this prod- 
uct multiplied by .7851 ^ives the solidity. 

Or, — 3. To the product of the diameters of the ends add 4 the 
square of the difference of the diameters, multiply the sum by .7854, 
and the product will be the mean area between the ends, which, mul- 
tiplied by the perpendicular height of the frustum, gives the solidity. 

Example. — The diameter of the larger end of a stick of round 
timber is 30 inches, that of the smaller end is 21 inches, and the 
lenijth of the stick is 3G feet ; required its contents. 

30 inches = 2^ feet, and 21 inches »s l| feet. 



By Rule 
By Rule 



\ 129 



^ 1.753-^2.5 vy^l.75 = 
cubic feet. Ans. 

2.5 X 1.75 « 4.375 + 

36=12). Ans. 



a3.6875X-^3^ X.7854 — 
.1875 = 4.5625 X .7854 X 



MENStJEATION OF SOLIDS. 



219 



Example. — The- interior diameter of the larger end 
of a circular cistern is 12 feet, that of the smaller end 
is 8 feet, and the perpendicular depth of the cistern is 
14 feet ; required its capacity in cubic feet. 
12 — 8 = 4X4 = 16 -r3 = 5.333 = J sq. of differ- 
ence of ends ; and 
12X 8-f- 5.333= 101.333 X •'7854 X 14= 1114.217. 

Note. — For an Example under Rule 1, see Mensuration of Pyramids, All rules which 
are applicable to Ihe measuring of a cone, or frustum thereof, are also applicable to the 
pyramid, or its frustum ; but, inasmuch as the areas of the ends of the two figures are 
not found with equal readiness, there V\ill usually be a choice in the employment of rules. 




OF SPHERES. 



To find the surface of a sphere or globe. 

Rule. — Multiply the diameter by the circumference ; or, multiply 
the square of the diameter by 3.1416, and the product will be the 



surface. 



To find ihe solidity of a sphere or globe. 



Rule. — Multiply the superficies by ^ of 
the diameter ; or, multiply the square of the 
diameter by 4 of the circumference ; or, mul- 
tiply the cube of the diameter by .5236, and 
the product is the solidity. 

Example. — Required the solidity of a 
cannon ball whose diameter is 9 inches. 




9 X 9 X 3.1416 — 254.46-[- sq. in., surface, or superficies, and 
254.46 X f = 381.7 cubic inches. Ans. 
Or, 9 X 9 X 9 = "29 X -5236 = 381.7. Ans. 



To find the convex surface of a spherical segment or zone. 

Rule.* — Multiply the height by the circumference of the sphere* 
Or, fi = 27rrv = tc (^r/"-[~ ^'')y ^^^ ^^^^ segment ; r being the radius 

of the sphere, d the diameter of the base of the segment, and v 

the height ; and 

p==:2^r/i = i7r(D-— fZ--[-4A2-f 8/iL'), for the zone; D and d 

being the diameters of the bases, h the height of the zone, and v 

the height of a spherical segment whose base is equal to the less 

base of the zone. 



220 MENSURATION OF SOLIDS. 



To find the solidity of a spherical segment. 

Rule. — 1. To the square of the height of the segment, add throe 
times the square of half the base, and multi- 
ply the sum by the height, multiplied by ^^^^^^^ 
•5230, and the product is the solidity. >^^^B^^^^ 

Or, — 2. From three times the axis of the ^^^H^l^^^^ft^ 

sphere, e /, subtract twice the height of the ^^KT * ^^^^k 

segment, and multiply the difference by the J^^L, JD^^I/ 

square of the height, multiplied by .5236, ^^B "^B^^f 

and the product will be the contents. JVp I^IC 

NoTB. — )i n di -\- b <fi. ^bcr=tf, and e/X ^ c — ^k J^ 

bd^= A/i^~d2 = ^ad. ^ ^ 

. Example. — The base, a </, of the segment a </ c, is lb inches, and 
the altitude, A r, is 8 inches ; required the solidity of the segment. 
8"^ + 9^X 3 =- 307 X 8 X -5236 = 1285.96 cubic inches. Ans 

To find the solidity of a spherical zone. 

Rule. — Add the stpiare of the radius of one base to the square of 
the radius of the other, and thereto add ^ the square of the height ; 
muliiply ihcir sum by the height, multiplied by 1.5708, and the prod- 
uct will be the contents. 

Example. — The base, g A, of the zone g h d a, is i feet, the base, 
a t/, is 3 feet, and the height, i b, is 2i feet ; the contents of the zone 
are required. 

1-^2 = 2, radius of base g h. I 2^i ^ q 25 — 3 = 2.0833 =« 
3 -^= 1-^ '' ** ** «^- I J square of height. Then, 
2'^ + 1.5^ + 2.083 = 8.333 X 2.5 X 1.5708 = 32.72+ cub. feet. 

Ans. 

To find the side of the greatest cube that can be cut from a given sphere. 
Rule. — Divide the square of the diameter of the sphere by 3, and 
the square root of the (juotient is the side ; or, multiply the diameter 
of the sphere by .57735, and the product is the side. 

Example. — The diameter of a globe is 15 inches; required the 
side of the greatest cube that may be cut from the globe. 
15 X 15 = 225 -i- 3 = V75 = 8.66 inches, or 
15 X .57735 = 8.66 inches. Ans. 



MENSURATION OF SOLIDS, 



221 




OF SPHEROIDS. 
To find the solidity of a spheroid. 
Rule. — Multiply the square of the 
revolving axis by the fixed axis multi- 
plied by .5236, and the product is the 
solidity. 

Example. — The fixed axis, a h, of 
the prolate spheroid ac b d, is 32 inches, 
and the revolving axis, c d, is 20 inches ; 
required the solid contents. 

20^ X ^^ X .5236 = 6702.08 cubic inches = 
6702.08 ^ 1728 = 3.88- cubic feet. Ans, 

To find the solidity of the segment of a spheroid. 
Rule. — When the base of the segment is parallel to the shorter axis 
of the spheroid. — From three times the length of the longer axis, 
subtract twice the height of the segment, and multiply the difference 
by the square of the height, multiplied by the square of the shorter 
axis, multiplied by .5236, and divide the product by the square of the 
longer axis ; the quotient will be the solid contents of the segment. 

Rule. — When the base of the segment is parallel to the longer axis 
of the spheroid. — From three times the length of the shorter axis 
subtract twice the height of the segment, and multiply the difiference 
by the square of the height multiplied by the longer axis, multiplied 
by .5236, and divide the product by the shorter axis ; the quotient 
will be the solidity. 

Example. — The longer axis, a b, of the spheroid a c b d, being 32 
inches, and the shorter, c d, 20 inches, what is the solidity of the seg- 
ment ef a, whose base, ef, is 12 inches, and height, g a, 4 inches? 

32 X 3 — 8 = 88X 16X20-X .5236=294891.52-^-1024 = 287.98 
cub. inches. Ans. 

To find the solidity of the middle frustum of a spheroid. 
Rule. — \Mien the ends of the frustum are parallel to the revolving 
axis. — To the square of the diameter of either end add twice the 
square of the revolving axis, and multiply 
the sum by the length of the frustum multi- 
plied by .2618 ; the product will be the so- 
lidity ; or the capacity, the frustum being a 
cask, and the measures taken of the interior. 

Example. — The diam.eters, cf and g h, 
of the frustum a d b c, arr 18 inches each, 
19* 




222 



MENSITRATION OF SOLIDS. 



the revolving axis, c </, is 23 inches, and the length of the frustum, 
a b, is 20 inches ; required the cubic capacity of the frustum. 
23^ X 2 + 18^= 1382 X 20 X .2618 = 7236.152 ~ 1728 = 4.187 
cubic feet. Ans, 

OF SPINDLES AND CONOIDS. 
To find the solidity of an elliptic spindle. 
Rule. — To the square of twice the diameter of the spindle, at i 
its length, add the square of its grcaiest diameter, and multiply the 
sum by the length, multiplied by .1309 ; the product will be the solid- 
ity, nearly. 

Example. — The greatest diameter, 
c d, of the elliptic spindle a c b d/i8 
12 inches, the diameter at J ilslenj^lh, 
ef, is 8i inches, and the len«Tih of the 
spindle, a b, is 40 inches ; required its solidify. 

X'lOX 




8.5 X 2 = 17, and IT-' + 12 ' 



.1309 



2267.188 cubic, in 



To find the solidity of a parabolic spindle. 
Rule. — Multiply the square 
of the middle diamotrr by the 
length of the spindle multiplied 
by .41888, and the product is 
the solidity. 

Example. — The diameter, 
c (/, at the middle of the ])ara- 
bolic spindle E d F r, is 15 
inches, and the length of the spindle, E F, is 40 inches ; required its 
solidity. 

15^ X 40 X .41888 :- 3769.92 cubic inches. Ans. 




To find the solidity of the middle frustum of a parabolic .spindle. 

Rule. — To eight times the square of the greatest diameter, add 
three times the scjuare of the least diameter, and four times the prod- 
uct of the two diainoters, and multiply the sum by the length of the 
frustum, multiplied by .05230, and the product is the solidity. 

Example. — The greatest diameter, c r/, of the frustum a c b d, is 
28 inches, the least diameter (that of either end) is 20 inches, and the 
length of the frustum, a Z>, is 40 inches ; required the solidity. 

28'^ X 8 + 20^ X 3 + 28 X 20 X 4 X 40 X .05236 =« ^^^Jg J 
cubic feet. Ans. 



a 



MENSURATION OF SOLIDS. 



223 



*ro find the solidity of a paraboloid, or parabolic 
conoid. 
Rule. — Multiply the area of the base by 
half the altitude ; or, multiply the square of 
the diameter of the base by the altitude mul- 
tiplied by .3927, and the product will be the 
solidity. 

Example. — The diameter of the base, cd, 
of the paraboloid, c ^ ^, is 40 inches, and its 
height,/^, is 40 inches ; required its solidity. 

40^ X w854 X 20 = 25132,8 cubic inches 




Ans. 



To find the solidity of a frustum of a paraboloid. 
Rule. — Add the squares of the diameters of the two bases 
together, and multiply the sum by the distance between. the bases, 
multiplied by .3927 ; the product will be the solidity. 

Example. — The diam.eter of the base, c c?, of the frustum, c dh a, 
is 40 inches, the diameter of the base, a b, is 22 inches, and the height 
of the frustum,/ e, is 26 inches ; required the solidity, or capacity. 

402 j^ 22- .= 2084 X 26 X -3927 = 21278 cubic inches. Ans. 

To find the solidity of a hypcrholoid or hyperbolic conoid. 
Rule. — To the square of half the 
diameter of the base add the square of 
the diameter midway between the base 
and vertex, and multiply the sum by 
the distance between the base and ver- 
tex, multiplied by .5236 ; the product 
will be the solidity. 

Example. — The diameter of the 
base, a b, of the hyperboloid, a bf is 
40 inches, the diameter midway between the base and vertex is 26 
inches, and the altitude of the figure, e /, is 24 inches ; required its 
solidity. 

20- + 26^ = 1076 X 24 X .5236 = 13521.4+ cubic in. A71S, 

To find the solidity of a frustum of a hyperboloid. 
Rule. — To the square of half the diameter of one end, add the 
square of half the diameter of the other end, and the square of the 
diameter midway between the two- ends, and multiply the sum by the 
length of the frustum, multiplied by .5236 ; the product is the so- 
lidity. 




^^^ MENSURATION OP SOLIDS. 

at the point equally dis.am from cither base s •" Tnche« II A T 
length of- the fn.tu,n, . r, is 18 inches ; required the «ohd!ty "' ''' 
20^+ 8.5^ + 29.P= 1319 X 18 X .5236 x= 12432 cub. in. Ans 

length of the fr»«lu,n dccreasej by Lm-'m,, noi. ^J' - ."-l""^"' »"1 niultiplied b, th. 
Instance, the last example- '^ ' niiiliiplii..J by .,%,4, g„ea the cubic contaoU. 

^" - '^ =|LX -55* = I2.6J + ,7 = 29.65 - „e» diameter, and 

To find the surface of a cyUndrical ring. 
Rule. —To the inner diameter of the 
nn(T add its thickness, and multiply the sum ^ 

by the thickness, multiplied by 9.8C96 : the 
product is the surface. 

ExAMPLK. — The inner diameter, A c, of a 
cylindrical rinp ia j.j i„^,p.,^ „„,, j,, j,,j^,j_ 

ncsa, ah, IS three inches ; required the super- 

12 + 3 = 15 X 3 X 9.8096 = 414 8q. incheT^S 
To find the xolidihj of a cylindncal rin^ 

n^^'TbtTiJ}'^ '"T *'',"r'^'' * '' ■' 12 inchw, and the thick- 
nesa, a ft^_is_3_inches; the solidity is required. 

12 + 3 X 3^ X 2.1674 = 333.1 cubic inches. Ans. 
♦For the CO cindcnl multipliers, and rule in deuil, kc OAro.s™ ,,.„,-. ,. 




MENSURATION OF SUPERFICIES. 



225 



OF THE KEGULAR BODIES. 

The Regular Bodies are five in number, viz. : 

The Tetrahedron or equilateral triangle ; a solid bounded by four 
equilateral triangles. 

The Hexahedron or cube ; a solid bounded by six equal squares. 

The Octrahedron ; a solid bounded by eight equilateral triangles. 

The Dodecahedron ; a solid bounded by tweke regular and equal 
pentagons. 

The Icosahedron ; a solid bounded by twenty equilateral triangles. 

The following Table shows the superficies and solidity of each of the 
regular bodies, ihe linear edge of each being 1. 



No. of sides. 


Names. 


Superficies. 


Solidities. 


4 

6 

8 

12 

20 


Tetrahedron. 

Hexahedron. 

Octahedron. 

Dodecahedron. 

Icosahedron. 


1.73205 
6.00000 
3.46410 
20.64573 
8.66025 


0.11785 
1.00000 
0.47140 
7.66312 
2.18169 



To find the superficies of any of the regular bodies, by help of the fore- 
going table. 

Rule. — Multiply the tabular number in the column of super- 
ficies by the square of the linear edge, and the product will be th^ 
surface. 

Example. — The linem^edge of a tetrahedron is 3 feet ; required 
the superficies. *^^^ 

1.732 X 32 = 15.588 square feet. Ans, 

To find the solidity of any of the regular bodies, by help of the fore- 
going table. 

Rule. — Multiply the tabular number in the column of solidities 
by the cube of the linear edge, and the product will be the solidity 
or contents. 

Example. — The linear edge of an octahedron is 2J feet ; required 
the solidity. 

.4714 X 2.5 X 2.5 X 2.5 = 7.3656 cubic feet. Ans 



1 



226 PROMISCUOUS EXAMPLES IN QBOMXTBT. 



PROMISCUOUS EXAMPLES IN GEOMETRY. 

I. To find the Diameter of a Circle, or the Side of a Square j whose 
Area shall bear a given ratio to the Area of a given Circle, or given 
Square. 

D =z diameter of given circle, or side of given square. 
d=z diameter of rofjuircd circle, or side of required squar* 
a-^b=z given ratio of area, d = \^(aJj^ -|- h). 

Example. — An engraver has a drawing which may be circum- 
scribed by a square whose sides arc 8 inches, and he wishes to cony 
it at J the size : required the si(K\s of the square that will cir- 
cumscribe the proix)sed copy. v/[(2 X 8-) -f-3J = G.532-in, Anit, 

2V? find the Sides of a Rectangle, Rhombus, or Rhondxnd, whose 
Area shall bear a given ratio to the Area of a given similar Rec- 
tangle, Rhombus, or Rhomboid. 

L z= length of given figure ; H = breadth or perpendicular 
height of given figure ; / = length of required figure ; h = breadth 
or perpendicular lieight of re(iulred figure ; a -7-6 ^ given ratio of 
area. 

l = ^{a L' -^ by. h = \f(a IV -^ /;) 

Or, L :i:: If :h, and H:h::L: 

ExAMTLK. — An engraver has a drawing which may i)e cjn iim- 
scribed by a rectangle whose length is 8, and breadth 6 ; and ho 
"wishes to copy it-at half the size: rerpiired the length and breadth 
of a rectangle that will eireumseribe the proposed copy. 
V/[(8 X ^)-7- 2] = 5. 65G85 = length of recpiired rectangle. ) . 
8 X 6 -^ 2 X 5.0568') = 4.242G1 = breadth required. ) ' 

11. A brace is to run 3 feet on the post of a building (from angle 
to extreme) and 1 feet on the plate. What must be the length of 
the brace (extreme from shoulder to shoulder), and at what an- 
gles must the shoulders of the tenons be cut ? 

y/(42 _^ 32>) __ 5 f^Q^^ length of brace. Ans. 

3 X ^6.14 -^(54-^) = 3G°.91, angle for head of brace. Ans. 

OQo — 36°.91 = 53<^.09, angle for foot of bface. Ans. 

in. . The roof of a building 30 feet wide is to have a height 
equal to one-third the width of the building ; required the length 
of the rafters, supposing the sides of the roof are to be equal, and 
the angles for their ends. 



PROMISCUOUS EXAMPLES IN GEOMETEY. 227 

Half width of buildings 15 feet ; height of roof = 10 feet. 
V/(152+ 10-) = 18.03 fg^i-^ length of rafters or slant height of 
roof. Ans. 

10X86.14-f-(18.03-|-7.5)=33°.74, pitch of roof, or angle 
for foot of rafters. Ans, 

90° — 33°. 74 = 56°.26, angle for head of rafters if they are to 
be bevelled together ; or (90° — 56°.26) X 2 z= 6 7°.48 (direct 
with that for the foot), if one is to rest upon the top of the other, 
or if they are to be halved together, or if an equilateral ridge-pole 
is to be used; in which latter case, the upper and lower angles of 
the pole must be 90°-|-56°.26 — 33°.74=112°.52 each, and the 
lateral angles 180° — 112°.52 = 67°.48 each. But if the upper 
ends of the rafters are to be squared, by which the pole will have 
two right angles, the lateral angles of the pole must be 90° each, 
the upper angle 112°.52, as before, and the lower angle 6 7°.48. 
Ans. 

IV. A king-post is to run 5.1 feet on the rafter, and 5 feet on 
the beam ; required the length of the post, and the angles for its 
ends, the perpendicular h (dropped from the extremity of the run 
on the rafter to the beam) being 2.8 feet. 

b= v/(5.l2— -2.8-) =4.26263 feet, distance from foot of per- 
pendicular to angle opposite post. 

p = v/[2.8' + (5 — &)'] = 2.89546 feet, length of post. Ans. 

86.14/i -f- (5.1 + ^b) = 33°.35, angle opposite post, or pitch of 

roof; and 90° — 33°.35 -f ^^-^^ X (o — 6) ^ ^^^^^^^ ^ ^^^ 

head of post. A7is. 

180° — 71°.44 — 33°.35 =:75°.21, angle for foot of post (in- 
verse to that for the head). Ans. 

The foregoing is applicable to a roof of equal or unequal sides, 
and to hip-roofs generally. 

Note. — Post perpendicular to rafter, acute angle of foot of post on out- 
side. 

Post perpendicular to beam, acute angle of head of post on outside. 

Post inclining toward opposite angle, and inner vertical angle greater than 
90°, acute angle of foot of post on outside ; of head of post on inside. 

Post inclining toward opposite angle, and inner vertical angle less than 90°, 
acute angle of foot of post on outside; of head, on outside. 

Post declining from opposite angle, acute angle of foot of post on inside; 
of head, on outside. 

V. Two perpendicular walls are standing on a plane 80 ft. apart ; 
one is 20 feet high, and the other 30 ; at what distance on the plane, 



228 PEOMISCUOUS EXAMPLES IN GEOMETRY. 

between them, from the base of the highest must a ladder be placed, 
so that, being inclined to either, it will reach the top ; and what 
must be the length of the ladder? 

80-' — 30-' ^ 202 = 5000 -^ (80 X 2) = 3GJ feet from base of 
highest wall. Ans. 

V (30 j'- + 30-) = 47.54 feet, length of ladder. Ans. 
Proof. 80 — 36^ = 43^ feet from base of lowest wall ; and 
V(43j^ + 202) = 47.54 feet, length of ladder, as before. 

A pole 100 feet in length is standing on a plane ; at what height 
must it be cut oiT, so that (the butt resting on the stump) the top 
will reach the ground 80 feet from the base of the stump ? 

803 
100 -f^ 

= 18 feet. Ans, 



= 18 feet. Ans. 



(D — //)II 1I</ 

VI. In a Cone, fr > = 11 ; tt = // ; 77 7 = D : 

D — a D 11 — h 

— = (I; D being the diameter of the base, H the altitude, 

d the diameter at any given altitude h above the base, and h tlio 
altitude above tlie base to any given diameter d, or II the slant 
length of the cone, and h the slant lenc^th of the frustum ; also, 
r>^* T. .. . (8 — 5)1)' II , , S(/^ ^ 

D — V g 

^ ^ = d ; S being the solidity of the cone, and s the solidity 

of the frustum. 

It is required to cut from a cone, by a section parallel to the base, 
a frustum containing 10 culnc foet ; the altitude (length of axis) of 
the cone is 14 foot, and tlio diameter of its base 4 feet ; what must bo 
the altitude of the frustum ? 

(58.6432— iro X<»4 14 

4-^^ -^^^ = .403 X -^ = 1.41 feet. A,^. 





2 




Or 


100- — 


80-' 


\Jly 


100 X 


2 


Cnno 


DA 





PROMISCUOUS EXAMPLES IN GEOMETRY. 229 

^nr. ^ . n (B — d)h (D — m)H , 
Vll . In a frustum of a cone, D Tt =m; — ^^ -j— = A ; 

D — m ' H — ■ h ^ h 

being the diameter of the greater base, d the diameter of the less 
base, H the altitude of the frustum, h the altitude at any given di- 
ameter m above the greater base, and m the diameter at any given 
(DTT DTT 

jy_^ — H)^^2618 + S = r^-—^\2^l^ = 

D'11.2618 

— pT -J- = solidity of cone completed from the frustum, S being the 

D^H.2618 D^H.2618 
solidity of the frustum ; and —j: -j- : — pr ^ s wW : m-^, 

or /s/ (D^ TT of-iQ "' ) = ^> the diameter of the less base of a frus- 
tum whose solidity is 5, cut by a plane parallel to the bases, 
from the greater end of the frustum whose solidity is S; also, 
/ s 3D'\ D f s 3m-\ m 

^V06T8-T-J-2=^'^Va:26T8-""4-;-2=^'^^^ 
s 

h, the altitude of the frustum, the diame- 



(D^ + m- + Dm).2618 

ter of whose less base is m, and solidity s. 

If the frustum whose solidity is 5 is to be taken from the smaller 

(D — </) X (S — 5) 
end of the given frustum, then /^D^ ^ tt ofiQ = ^> 

* (D— m)H 
S being the solidity of the given frustum ; and H y. . = A , 

altitude of frustum whose solidity is 5, taken from smaller end of 
given frustum. 



^A._(;^^J = H, and ^H=+ (^7 = A, A be- 

ing the slant length of the frustum. 

A round stick of timber has diameter of greater end 3 feet, diame- 
ter of less end 2 feet, and length of axis 12 feet ; at what distance on 
the axis, from the greater end, must I cut this stick, by a section 
parallel to the ends, in order that the frustum, cut from the greater 
end, may contain 25 cubic feet? 

(3 — 2)25 \ 12 

.3 -^(33- ^-4.^X3:32 = 3-955 feet. Ans. 

The interior dimensions of a vessel in the form of a frustum of a 
cone are, depth 12 inches, bottom diameter 10 inches, top diameter 
20 



2ii{) PROMISCUOUS EXAMPLES IN GEOMETRY. 

6 inches ; to what depth must the vessel be filled, it standing level 
upon its bottom, in order that it may contain I4 wine gallons ? 

(10__6) X 231 X 1.5\ 
^(103 __ .^ 12 X ^^018 ) "^ ^-^^^ "^ diameter at sur- 
face of contents, and 

(10 — 8.23G4) X 12 



10 — G 



=x 5.2908 inches. Ans. 



\T!II. The perpendicular depth (H) of a vessel in the form of a 
frustum of a cone is 7 feet, the bottom diameter (D) 5 feet, and the 
diameter of the open mouth, or top {(I) 3 foot ; the vessel is turned 
on edge, and fillod with water till the level suriiice of the watvr just 
touches the upprTinuJst point in tlie bottom and lowermost in the top; 
refjuired the quantity of water in thw vessel. 

j ^ZSd X -2018011 « content, then 

5 X '5 — 3 X V (-^ X 3 = 13.3S1 ^ (5 — 3) = C.GO X T X 5 X .2018 
= 01.3 cubic feet. Ans. 

The same vessel, and D the open mouth, other things as before, 

D V(Dr/)— rf^ 

iTZTJ X .2018</II = contents = 28.49 cubic feet. Ans. 

IX. A vessel 12 feet liigh (II), and kept constantly full of water, 
has an opening in its side 4 feet (h) above the bottom, from which a 
jet is projected ; to what disUinco on the plane, level with the bottom, 
is the jet projected ? 

\/ (II — /i)A X 2 = projection, then 

V'(T2^=^4)~X 4== 5.057X2= 11.314 feet. Ans. 



TMGONOMJBTRY, 231 



TRIGONOMETRY. 



Trtgonometry is a branch of Geometry. It treats of the relations 
of the sides and angles of triangles to each other. It enables us, 
having any three of the three sides and three angles of a triangle 
given, and one of them a side, to find the rest. It treats of the 
mensuration of the angles of triangles, therefore, and of the mensu- 
ration of the sides. 

Trigonometry is of two kinds, Rectilinear and Spherical. 

The former treats of right-lined triangles, and the latter of trian- 
gles formed by the intersections of three great circles upon the sur- 
face of a sphere. 

Rectilinear trigonometry is often denominated plane trigonometry^ 
and is divided into two parts, rectangular and oblique-angular. 

In rectilinear trigonometry, instead of making use of an arc of a 
circle as the measure of an angle, the sine, tangent, secant, or cosine, 
cotangent, cosecant, of that arc, is used ; and the sides of plane tri- 
angles, therefore, take these names, interchangeably, according as 
one or another is made radius of the arc supposed, and according as 
one extremity or the other of the radius is made centre. 

The circumference of every circle is divided into 3G0 equal parts, 
called deo;rees, (^) ; each degree into 60 equal parts, called minutes, 
(') ; and each minute into GO equal parts, called seconds, {" ) : 
and an angle at the centre of a circle, formed by any two sides of a 
triangle, is as many degrees, minutes and seconds, as there are de- 
grees, minutes and seconds, in that portion of the circumference that 
the sides forming that angle embrace, or may be supposed to enclose. 

And because the sides of triangles are sines, tangents, etc., of the 
arcs that measure the angles, they are also said to be sines, tangents, 
^c, of the angles that are measured by those arcs. (See Geometry 
— definitions.) 

Proposition 1. — In every right-angled triangle, if the hypotenuse 
be made radius, one of the legs will be the sine of the angle whose 
vertex is made centre, and the other will be the cosine of the same 
angle, and either extremity of the hypotenuse, or the vertex of either 
acute-angle of a right-angled triangle may be made centre. Thus, 
if, in the diagram ABC, annexed, the hypotenuse A C be made 
radius, and A the centre, B C will be the sine of the angle at A, 
or of the angle A, and A B will be the cosine of the same angle. And 
if A C be made radius and C the centre, A B will be the sine of the 
angle C, and B C will be the cosine of the same angle. 

Either leg, therefore, of a right-angled triangle that is the sine 




282 TMOONOMETP.Y. 

of one of the acnte-angles, is also the cosine of the other acute- 
angle ; and the legs of a right-angled triangle, the hypotenuse being 
raSus, are sines oi their opposite angles. 

Proposition 2. — In every rights 
angled triangle^ if one of the legs bo 
made radius, the other will be the 
tangent of the acute-angle whose 
vertex is made centre, and the hy- 
potenuse will be the secant of tfi 
same angle, and either leg may )>• 
made radius. Thus, if A B be niiul' 
radius, A will be centre, BC will h 
the tangent of the angle A, and th« 
'••...^ hypotenuse will be the secant of th^ 

'*--....^ same angle. And if B C be mad 

radius, C will be centre, A B will !.• 
the tangent of the angle C, and A C 
will be the secant of C. 

Either leg, therefore, of a right-angled triangle that is the tangent 
of one of the acute-angles, is also the cotangent of the otlier ; and 
the hypotenuse being secant of the acute-angle whose vertex is ma»ie 
centre, is also cosecant of tlie other ; one of tlie legs of a right-angled 
triangle being made radius, the other is the' tangent of its opposite 
angle. 

As four right-angles can be formed about the Bame point, there- 
fore every right-angle is equal to a quadrant of the circle, or 9(P. 

The three angles of any triangle are equal to two right-angles, 
or 180^ 

The two acute-angles of a right-angled triangle are equal to a 
right-angle. 

When a particular angle of a triangle is referred to, and the 
three letters at the three angles of that triangle are used to express 
it, the letter at the angle referred to is placed in the middle ; thus, 
both the angle and the triangle are indicated : in the expression, 
■ The angle A B C\ the angle B, in the triangle A B C, is meant. 
In the annexed tal)le of natural sines, C(>sines and tangents, to given 
angles, the numbers, though not so marked, are considered as deci- 
mals to radius 1. 

The natural sine of 90^, tlierefore, is 1 ; and the natural cosine 
of 90^ is 0. 

The tangent of 45^ is 1, and so is the cotangent. 
The tangent of 90^ is infinite. 

If two antjles of a triangle are given, the other is said also to be 
given, for it is the difference between the sum of the two and 180"^-. 
If one of the acute-angles of a right-angled triangle be given, all 
the angles of that triangle are said to be given, for the sum of the 



TRIGONOMETRY. 



233 



three angles of any triangle, minus the sum of any two, is equal the 
third, and the right-angle is known for 90^. 

In trigonometrical expressions and formulas, the following abbre- 
viations and contractions are often made use of, yiz. : 

R, for tabular Radius, natural or logarithmic sine of 90^, or of 
the right-angle, (1.)* 

sin A, {or any other letter,) for sine of the angle A, or whatever 
other letter. 

cos A, for cosine of the angle A. 

tan A, for tangent of the angle A. 

sec A, for secant of the angle A. 

cosec A, for cosecant of the angle A. 

cot A, for cotangent of the angle A, 

coversin, for coversed sine, 

sin A comp B, for sine of the angle A, the complement of which 
angle is the angle B. That is, the angle A subtracted from 90^, the 
difference will be the angle B. 

These expressions, such as sin A, tan B, &c., or their equivalents, 
when used as terms in the statement of a problem, or its solution, 
are to be taken as referring to the natural or tabular sine, tangent, 
&c., of the angle indicated, and not to the sides taking those names. 
Sometimes the contraction nat. or tab. is prefixed, (as nat cos B,) 
but usually it is omitted. 

In any right-angled plane triangle, ABC — 

A being one of the acute angles, and C the other ; 



sin A = cos C 
tan A = cot C 
sec A = cosec C 



cos A = sin C. 
cot A = tan C. 
cosec A = sec C. 



And of the game angle, A or C, considered separately, 



tan 


X cos = sin 


cot 


X sin 


= cos 


cos 


-^ cot = sin 


sin 


— tan 


= cos 


tan 


-^ sec = sin 


cot 


-^ cosec = cos 


R(l) 


-^ cosec = sin 


R 


-^sec 


= cos 


sin 


X sec = tan 


cosec 


X cos 


= cot 


sin 


-f- cos = tan 


cos 


-r- sin 


= cot 


sec 


-1- cosec = tan 


cosec 


-^sec 


== cot 


R 


-^ cot = tan 


R 


-f-tan 


= cot 


cosec 


X tan = sec 


cot 


Xsec 


= cosec 


tan 


-^ sin = see 


sec 


-r- tan 


= cosec 


cosec 


-1- cot = sec 


cot 


-^ cos 


= cosec 


R 


-^- cos = sec 


R 


-f- sin 


= cosec 



* The logaritlimic liulius or tiil)ii1ar sine of 90^ is 10 : the natural radius or tabular sine 
of 90^ is 1. A table of lagarithmic sines, tangents, &c., is called a table of artificial sines, 
tangents, &c. 90 * 



234 



TBIGONOMETRT. 




perp. 
hyp. 



hyp. 

pen>' 

baso 
baso 
perp. 
hyp. 



base 
hyp. 
perp. 



In any right Angled plane triangle, 
A B C, (Fig.) 

B C -^ A C (tan -r- sec) = sin A. 
B C -r- A B (sin -|- cos) = tan A. 
A C -f- A B (R ~ cos) = sec A. 
A B -i- A C (tan -f- sec) = sin C. 
A B -f- B C (sin -7- cos) = tan C. 
A C -i- B C (R -h cos) = see C. 

Or, if we denominate the longest 
side of the triangle hypotenuse^ and 
of the other two sides make one 
base and the other 'perpendicular; 
then — 



: nat sin of angle opposite perp., or nat cos of angle opposite base. 
: nat sin of angle opposite base, or nat cos of angle opposite perp. 
: nat tan of angle opposite perp., or nat cot of angle opposite baso. 
: nat tan of angle opposite base, or nat cot of angle opposite perp. 
: nat sec of angle opposite perp., or nat coscc of angle opposite base. 
: nat SCO of angle opposite baae, or nat coscc of angle opposite perp. 



With reference to the sides of right-angled triangles, considered 
in connection with the natural sines, tangents, &c., of tlieir opp'>^T^ 
and included angles : — 



Preceding Fig. 

A C X sin A = B C. 
A B X tan A = B C. 
A B X sec A = A C. 
B C X cosec A = A C. 
A C X cos A = A B. 
B C X cot A = A B. 



A C -T- sec A = A B. 
A C -f- cosec A = B C. 
AB-^cotA =BC. 
A B -r- cos A = A C. 
B C -T- sin A = A C. 
B C ~ tan A = A B. 



In eyery plane triangle, — right-angled, acute-angled or obtuse- 
angled, — the natural smes of the angles are to each other as the 
opposite sides. 

Thus, in the triangle ABC, annexed. 



TRIGONOMETRY, 



235 



sin A : sin B : : B C : A 0. 
sin B : A C : : sin C : A B. 
B C : A B : : sin A : sin C. 

In every plane triangle, the 
sum of the squares of the sides 
adjacent any angle, minus the 
square of the side opposite that ^ 
angle, divided by twice the 
product of the sides adjacent, equals the natural cosine of that 
angle. 




Thus, 



AB^ + AC'— Ba 
A B X ^rij' X 2 

AB'^-f BC'^ — AC - 
2^BX B C 

AC2-|-BT^-_AB' 
2AC X BC 



= cos A. 



= cos B. 



= cos C. 



Or, if the sides of the triangle be of unequal lengths, and we de- 
nominate them according to their relative lengths, as longest, medi- 
ate, shortest, we have the following more distinctive formulas: 

longest^ -\~ med- — short- 

longest XmedX2~ = ^^^ ^^^ ^^^^^^^ ^^S^^' 



= nat cos mediate angle. 



longest- -}" short- — med- 

long X short X 2 
med'-^ -\- shortest^ — longest^ 

mediate X shortest X 2 = °^' °°^ g'^**^^' ^°S^«- 
V (1 — sin-) of any angle = cos of that angle. 
1 — (2 X sin2) of i any angle = cos of that angle. 
That is, A being the angle — 

1 — 2sin2ofi A = cosA. 

In every plane triangle, as the sum of any two sides is to their 
difference, so is the natural tangent of half the sum of the angles 
opposite those sides to the natural tangent of half their difference ; 
and half the sum of the two angles, plus half their difference, equals 
the greater of the two, and half the sum of the two angles, minus 



236 



TRIGONOMETRY, 



half their di^renco equals the less, the greater angle being op- 
posite the longer side. 

Thus, in the oblique-angled triangle, ABC, (Fig.) 

AC + BC:AC^BC::tan ^L±l , ^^ ^ ^^ 

A + B AurB or 

^ + —^ = B, and ^" 

A^B 

= A. 

AB + AC:AB — AC 

tan i(C + B) :tau 4(C — B),^^ 
and 

4(C + B) + i(C — B) =.C, and4(C + B) — 4(C--B) = B. 

A B 4- b'c : A B — B C : : tan 4(C + A) : tan 4(C — A), and 

C -f A C — A . C 4- A C — A 





A C + A 

— = C, and — 2 — ' 



2 



= A. 



SOLUTIONS, — RlGHT-ANOLED TRIANGLES. 




ABC, the triangle, 
A C, hypotenuse, 

A B and B C, legs ; B, the right 
angle. 

The hypotenuse at id angles given, to 
find the legs. 

Suppose A C 51 feet, A 28^ KY, Jsj 
V (consequently) 00^ — 28^ KY = 

ci° 50^ 

The sines of angles are to each ''*••.... 

other as the sides opposite those ' -^ 

angles. 

Turning, therefore, to the tables of natural sines, &c., we find, in 
the column of sines, against angle 28^ 10', sine .47204, consequently — 

R AC sinA BC 

1 : 51 :: .47204 : 24.1 feet. 

Having now A C and B C, side A B may 1)0 found by the rules in 
geometry ; or, turning again to angle 28^ 10', in the tables, we find, 



TEIGONOMETRY. 



23T 



against that angle, cosine .88158* ; and as the cosine of one of the 
acute angles of a right-angled triangle is the sine of the other acute 
angle, consequently, 

R AC sinC AB, 
1 : 51 : : .88158 : 45 feet. 

The angles and one leg given, tQ find the hypotenuse and other leg. 

As before, C = 61° 50', and A = 
90° — 6P 50^ = 28° 10'; AB = 
45 feet. 



sin A B R 

.88158 : 45 : : 1 

sin C A B 

.88158 : 45 

Or, R A B tan A 
1 : 45 : : ,53545 



AC 
51 



sin A 
.47204 



BC 
24.1. 



R 

1 



AB 
45 : 



sec A 
1.13433 



BC 
; 24.1. 

AC 
51, 



A,^ 





K 


/\ 


\ 


/ \ 


\ 


X \ 


\ 




\ 


/ ■\^ i 


\ 


/ 


, 



The hypotenuse and one leg given, to find the angles and other leg. 
Let A C = 85 and B C = 58 ; then — 



AC R 

85 : 1 : ; 



BC 

58 : 



sin A 

.68235. 



Turning now to the tables of nat. sines, &c,, we find, in the col- 
umn of smes, against the number 68235, or the number nearest 
thereto, angle 43°. We therefore have — 

A C : R : : B C : sin A, 43° ; and 

90° — 43° = 47° = angle C. 

R A C sin C A B 
1 : 85 : : .73135 : 62.16. 

The legs given, to find the angles and hypotenuse. 
Let A B = 54.7, and B C = 32 : 

. A B R B C tan A 

54.7 : 1 : : 32 : : .58501. 

Turning now to the tables of natural sines, tangents, &c., we find 
in the column of tangents 58501, or a number near thereto, and 

* When an angle is greater than 45^ the sine, tangent, &c., of its complement is used. 



238 



TRIGONaMETRY. 



against that number we fiiiil 30° 20' ; the angle A, therefore, ig 3(F 
20', and the angle C is 

90° — 30° 20' = 50^ 40'. 
sin A BC R AC 
.50503 : 32 : : 1 : G3.3G. 
Or, sin 90° : B C : : sec 59^ 40' : A C 
1 32# 1.98008 63.36. 



•1 



SOLUTIONS, — Obuqui-angled triangles. 

Let A B C be the triangle : (Fig. annexed.) 

The angles and one side given ^ to find tJie other sides. 

Suppose A 20° 40', B 56°, C 
180- — (20° 40' + 56°) = 97° 
20', and A C 40 feet. 

sin B : A C : : sin A : B C. 




The an^lc B is greater thiv 
45°; its sine, therefore, is tli 
,jg cosine of its complement, or th*- 
cosine of wliat it lacks of 90 . 
Its sine, therefore, is tlic cosine of 90° — • 50^ = 34^. Turning n u 
to 34° in tlic tabh^s, we find, against that angle, cos .82904 ; .^J.'it4, 
consequently, is the sine of 56°, and 

sin B AC sin A B C 
.82904 : 40 : : .44880 : 21.65 feet, 
sin B : A C : : sin C : A B. 
The angle C is not only greater than 45^, but it is greater than 
90°. Its sine, therefore, is the cosine of the difTerenco of its suppl. 
ment and 90^. It is the cosine of the difference of 180° — 79 L'< 
= 82° 40' and 90^ = 7^ 20'. As in the proceding case, its sin- 
therefore, is the cosine of the difference between itself and 90°. 

Turning now to anp;le 7^ 20', in the tables, we find its cosiiv 
.99182 ; therefore we liave — 

* sin B AC sin C A B 

.82904 : 40 : : .99182 : 47.85 feet. 

Two sides and an angle opposite to one of them giren^ to find the other 
angles, and otlier side. 

As before, A B 47.85, B C 21.05, C 97° 20'. 

A B : sin C : : B C : sin A, 26° 40' ; and 
180° — (97^ 20' + 26° 40') = 50^ = B. 



TRIGONOMETET. 



239 



sin € : A B : : sin B : A C ; or, 
sin A : B : : sin B : A C. 

Tivo sides and their contained angle given, to find the other angles 
and side. * 

Suppose A B 47.85, B C 21,65, B 56^ 

A B + B C : A B ^ B C : : tan J(A -f C) : tan J(A o^ C). 

^ ISQo^B ^ R (1) 

tan 4(A + C) = tan ^ tan 90°^ 4(180^ — B). 

tan 4(180^ — B) = cot 90^ ^ J(180^ — ^) ; therefore, 
A B + B A B ^ B C tan 62^ tan 

69.5 : 26,2 :: 1.88073 : .70899, 35° 2(y; and 

^(A + C) 4(A^C) C 4(A + C) 4(AunC) A 

62^ + 35.20 =97^20' and 62^ —35^20' =26^40', 

\ sin A : B C : sin B : A C, or 

sin C : A B : : sin B : A C, 



(The three sides given, to find the 

I angles, 

\ 

I ■ Let A B C be the triangle. 

AB47.85, AO40, BC21.65. 



I AB- + AC^— BC^-f-ABX ACX2 = cos A. 

|. 3420.9 3828 .89365,26^^40'. 

' B C : sin A : : A C : sin B, 

1 21.65 .44880 40 .82919 = cos comp B, = 90° — 34° = 56<^ = B, 

j 180° — 26° 40' + 56° = 97° -20^ = C, 

Or, AB^ + BC^ -^AC'-T- A B X B C X 2= cosB 

J 1158,34 2071.9 .55907 = sin comp B = 

\ 90° — 34° =^56° = B. 




|Or,4(AB+-AC4-BC)XBC^J(BA + AC + BC)-T-ABXAO 

j = cos^ 4^. 

54,75 X 33.1 ~ 1914 = V.94682=.97305 = cos4A,13°2a'; and 
13° 20' X 2 = 26° 40' = A. 



240 TRIQOTfOMETRT. 

2B 

cos 2- 

ab+a^c-;bc^^^^ab4-ac+bc^^^^-^^ 

C0»2 J C. 

In general, however, when the three sides of a triangle are j^ivon. 
and the angles are required, it is customary to drop a pcrpondit ular 
upon the longest side of the triangle, or one that will fall within tli- 
figure, whereby the triangle is divided into two right-angled tri 
angles; and then to find the angles, hy the rules fur right-angled 
triangles. In this case the sum of the two vertical angles will be 
€(jual the angle at the vertex. For rules for dropping the perpeo- 
dicular, see Gkoxetry — Triangles, 

To find the angles of a right-anffled triangle approximatebj hy means 
of the sides y or without the aid of trigonometrical tobies. 

Let r represent the hypotenuse, 9 the shortest side, c the longest 
side, A the smallest angle, or angle opposite the shortest side. 

86.139615 
^;rTirT"Try,vei7 nearly.* 

ExAMPLK. — ^Vhat is the smaller of the two acvte angles of a 

right-angled triangle whose hypotenuse is 100, shortest side 48.7S5, 
and longetjt side 87.321 V 

86.13961 X 48.735 __ 
100 + 43.6605 + 0.3654-^^ .148, or 29 8 53 , 
true angle = 29° 10'. Ans. 
90O — 29^.148 z= 60^.853 or 60° 51' 7" = greater acute angle. 

* Sec CmcLE> length of arc of» Ac 



TRIGONOMETRY. 



241 



TABLE OF NATURAL SINES, COSINES, AND TANGENTS. 



D. M. 


Sine. 


Co^ne. 


Tangent. 


D. M. 


Sine. 


Cosine. 


Tangent. 


1 


00029 


10000 


00029 


7 


12187 


99255 


12278 


6 


00145 


10000 


00145 


10 


12476 


99219 


12574 


10 


00291 


10000 


00291 


20 


12764 


99182 


12869 


20 


00582 


99998 


00582 


30 


13053 


99144 


13165 


30 


00873 


99996 


00873 


40 


13341 


99106 


13461 


40 


01164 


99993 


01164 


50 


13629 


99067 


13758 


50 


01454 


99989 


01455 


8 


13917 


99027 


14054 


1 


01745 


999&5 


01745 


10 


14205 


98986 


14351 


10 


02036 


99979 


02036 


20 


14493 


98944 


14648 


20 


02327 


99973 


02328 


30 


14781 


98902 


14945 


30 


02618 


99966 


02619 


40 


15069 


98858 


15243 


40 


02908 


99958 


02910 


50 


15356 


98814 


15540 


50 


03199 


99949 


03201 


9 


15643 


98769 


15838 


2 


03490 


99939 


03492 


10 


15931 


98723 


16137 


10 


03781 


99929 


03783 


20 


16218 


98676 


16435 


20 


04071 


99917 


04075 


30 


16505 


98629 


16734 


30 


04362 


99905 


04366 


40 


16792 


98580 


17033 


40 


04653 


99982 


04658 


50 


17078 


98531 


17333 


50 


04943 


99878 


04949 


10 


17365 


98481 


17633 


3 


05234 


99863 


05241 


10 


17651 


98430 


17933 


10 


05524 


99847 


05533 


20 


17937 


98378 


18233 


20 


05814 


99831 


05824 


30 


18224 


98325 


18534 


30 


06105 


99813 


06116 


40 


18509 


98272 


18835 


40 


06395 


99795 


06408 


50 


18795 


98218 


19136 


50 


06685 


99776 


06700 


11 


19081 


98163 


19438 


4 


06976 


99756 


0(')1)'.)3 


10 


19366 


98107 


19740 


10 


07266 


99736 


07285 


20 


19652 


98050 


20042 


20 


07556 


99714 


07578 


30 


19937 


97992 


20345 


30. 


07846 


99692 


07870 


40 


20222 


97934 


20648 


40 


081-36 


99668 


08163 


50 


20507 


97875 


20952 


50 


08426 


99644 


08456 


12 


20791 


97815 


21256 


5 


08716 


99619 


08749 


10 


21076 


97754 


21560 


10 


09005 


99594 


09042 


20 


21360 


97692 


21864 


20 


09295 


99567 


09835 


30 


21644 


97630 


22169 


30 


09585 


99540 


09629 


40 


21928 


97566 


22475 


40 


09874 


99511 


09923 


50 


22212 


97502 


22781 


50 


10164 


99482 


10216 


13 


22495 


97437 


23087 


6 


10453 


99452 


10510 


10 


22778 


97371 


23398 


10 


10742 


99421 


10805 


20 


23062 


97304 


23700 


20 


11031 


99390 


11099 


30 


23345 


97237 


24008 


30 


11320 


99357 


11394 


40 


23627 


97169 


24316 


40 


11609 


99324 


11688 


50 


23910 


97100 


24624 


50 


11898 


99290 


11983 











21 



242 



XSIGONOMETBT. 



D. M. 


Sine. 


Cosine. 


Tanpent. 


D. M. 


Sine. 

87191 


Cosine. 


Tanpent. 


14 


24192 


97080 


24983 


21 60 


92827 


4(XH35 


10 


24474 


90959 


25242 


22 


87461 


92718 


40403 


20 


24756 


90887 


25552 


10 


87730 


92609 


40741 


30 


25038 


96815 


25862 


20 


87999 


92499 


41081 


40 


25820 


96742 


26172 


80 


88268 


92888 


41421 


60 


26601 


96607 


26483 


40 


88537 


92276 


41703 


16 


25882 


96598 


26795 


60 


88805 


92164 


42105 


10 


26163 


96517 


27107 


28 


89078 


92060 


42447 


20 


26448 


96440 


27419 


10 


89841 


91936 


427'»1 


30 


20724 


96363 


27782 


20 


39608 


91822 


\ 


40 


27(X)4 


9628;5 


28040 


80 


89876 


91706 


\ 


60 


27284 


96206 


28300 


40 


40141 


9161K) 


4uMib 


16 


27604 


96126 


28075 


60 


40408 


91472 


44175 


10 


27848 


96046 


289'JO 


24 


40674 


91865 


■\r-- 


20 


28123 


95904 


29305 


10 


40989 


91236 


I 


80 


28402 


96882 


2'.C,21 


20 


41204 


91116 


1 


40 


28680 


96799 


2*.n»3H 


80 


41469 


909<»Kj 


•1 


60 


28960 


95715 


80255 


40 


41784 


90876 


4;..:.i , 


17 


29237 


95630 


80573 


60 


41998 


90768 


40277 


10 


29516 


95545 


80891 


26 


42262 


90681 


4or.m 1 


20 


29793 


95459 


81210 


10 


42626 


90607 


1 


30 


30071 


95872 


81530 


20 


42788 


90888 


•1, 


40 


30348 


96284 


31850 


80 


43061 


90259 


47698 


60 


30025 


95195 


32171 


40 


48818 


90133 


48066 


18 


80902 


95106 


32492 


60 


48676 


90007 


48414 


10 


81178 


95015 


82814 


26 


48837 


89879 


48778 


20 


814r>4 


94924 


33130 


10 i 44098 


89752 


49134 


80 


31730 


'>i832 


33400 


20 


44869 


89623 


49496 


40 


82006 


94740 


83783 


80 


44620 


89498 


4'.»s:)8 


60 


32282 


94646 


84108 


40 


44880 


89363 


50L"J2 


19 


32557 


94552 


34433 


60 


46140 ' 89232 


;'< ' ' ^ ~ 


10 


32832 


94457 


34758 


27 


453"«'» 


80101 




20 


83106 


94361 


85085 


10 


46<>58 


88968 




80 


33381 


94204 


35412 


20 


46917 


88835 


.') 1 - - 


40 


33055 


94107 


35740 


80 


46176 


88701 


5l'' '■ ' / 


60 


83929 


940(]8 


30008 


40 


46483 


88566 


52427 


20 


84202 


93909 


30397 


60 


46690 


88481 


62798 


10 


84475 


93809 


36727 


28 


46947 


88295 


58171 


20 


34748 


937r.9 


37057 


10 


47204 


88168 


53545 


30 


35021 


93007 


37388 


20 


47460 


88020 


53920 


40 


85293 


93505 


87720 


30 


47716 


87882 


64296 


60 


35565 


98402 


38053 


40 


47971 


87743 


M073 


21 


85837 


93358 


88386 


60 


48226 


87603 


65051 


10 


36108 


93253 


38721 


29 


48481 


87462 


55431 


20 


36379 


93148 


39055 


10 


48786 


87321 


55812 


30 


36650 


93042 


39391 


20 


48989 


87178 


56194 


40 


30921 


92935 


39727 


30 


49242 


87036 


66577 



TEIG0N0METR7. 



243 



D. M. 


Sine. 


Cosine. 


Tangent. 


D. M. 


Sine. 
60876 


Cosine. 
79335 


Tangent. 


29 40 


49496 


86892 


67962 


37 30 


76733 


60 


49748 


86748 


67348 


40 


61107 


79158 


77196 


30 


60000 


86603 


57735 


60 


61337 


78980 


77661 


10 


60262 


86467 


68124 


38 


61566 


78801 


78129 


20 


60603 


86310 


68513 


10 


61795 


78622 


78698 


30 


60754 


86163 


58904 


20 


62024 


78442 


79070 


40 


61004 


86015 


69297 


30 


62251 


78261 


79644 


50 


51264 


85866 


59691 


40 


62479 


78079 


80020 


31 


61504 


86717 


60086 


50 


62706 


77897 


80498 


10 


61753 


85567 


60483 


39 


62932 


77716 


80978 


20 


52002 


86416 


60881 


10 


63158 


77531 


81461 


30 


62250 


85264 


61280 


20 


63383 


77347 


81946 


40 


52498 


85112 


61681 


30 


63608 


77162 


82434 


60 


62746 


84959 


62083 


40 


63832 


76977 


82923 


82 


62992 


84805 


62487 


60 


64056 


76791 


83416 


10 


53238 


84650 


62892 


40 


64279 


76604 


83910 


20 


63484 


84495 


63299 


10 


64501 


76417 


84407 


30 


63730 


84339 


63707 


20 


64723 


76229 


84906 


40 


53975 


84182 


64117 


30 


64945 


76041 


86408 


60 


64220 


84026 


64528 


40 


65166 


76851 


86912 


33 


54464 


83867 


64941 


60 


65386 


75661 


86419 


10 


54708 


83708 


65355 


41 


65606 


75471 


86929 


20' 


64951 


83549 


66771 


10 


65825 


75280 


87441 


30 


55194 


83389 


66189 


20 


66044 


75088 


87966 


40 


66436 


83228 


66608 


30 


66262 


74896 


88473 


60 


66678 


83066 


67028 


40 


66480 


74703 


88992 


34 


66919 


82904 


67451 


60 


66697 


74509 


89515 


10 


56160 


82741 


67876 


42 


66913 


74314 


90040 


20 


56401 


82577 


68301 


10 


67129^ 


74120 


90569 


30 


56641 


82413 


68728 


20 


67344 


73924 


91099 


40 


56880 


82248 


69157 


30 


67559 


73728 


91633 


60 


57119 


82082 


69588 


40 


67773 


73581 


92170 


36 


67358 


81915 


70021 


60 


67987 


73333 


92709 


10 


57596 


81748 


70455 


43 


68200 


73135 


93252 


20 


67833 


81580 


70891 


10 


68412 


72937 


93797 


30 


68070 


81412 


71329 


20 


68624 


72737 


94345 


40 


68307 


81242 


71769 


30 


68835 


72537 


94896 


60 


68543 


81072 


72211 


40 


69046 


72337 


95451 


36 


58779 


80902 


72654 


60 


69256 


72136 


96008 


10 


59014 


80730 


73100 


44 


69466 


71934 


96569 


20 


59248 


80558 


73547 


10 


69675 


71732 


97133 


30 


59482 


80386 


73996 


20 


69883 


71529 


97700 


40 


59716 


80212 


74447 


30 


70091 


71325 


98270 


50 


59949 


80038 


74900 


40 


70298 


71121 


98843 


87 


60182 


79864 


75355 


60 


70505 


70916 


99420 


10 


60414 


79688 


75812 


45 


70711 


70711 


1. 


20 


60645 


79512 


76272 











244 TRIGONOMETRY. 

Note. — The foregoing Tables present sines, cosines, ami tangents, calculated for every 
degree and ten minutes of the quadrant. To furnish tables calculated for a smaller divis- 
ion of the circle than ten minutes, as for five minutes, or for one minute, wnuM occupy too 
much space in this work. Besides, calculations to the sixth of i il- 

eal purposes, are sufficiently minute. If, however, greater pn- 1 a 

furnish a very simple means whereby to obtain it, and to ali;. f 

scientific minuteness and accuracy. 

Suppose, for instance, in the solution of a problem, the sine .64380 appears : now, on 
turning to the tables, the nearest sine we find to this is .04279, the sine of 4(P ; while the 
next nearest is .64501, the sine of 4<P 10' ; now, .64279 -f .64501 = 1.28780 -7- 2 = .64390 
= sin 40^ 5' -, .64390, therefore, is the sine of an angle a tritle less tliau 4lP 6', but nearer 
to that angle than t<j any other of full minutes. 

Again, suppose the sine .51433 appears ; the nearest sine in the tables to this is .51504, 
wliicli is the sine of 3P ; and the next nearest ijj .51254, the sine of 30^ 50* ; .51504 -\ 
.51254 = 1.02758 -^ 2 = .51379 = sin 30^ 55', and .61504 -f .51379 = 1.02883 -7- 2 = 
.51442 = sin 30^ 57]' ; .514:33, therefore, is the sine of 30^ 57', very nearly. 

The foregoing principles are also ai)pliaiblc U) the cosines and tangents, and in tha 
satne manner. 

The versed sine of any angle = 1 — cosine of that angle. 
The covcrscd sine of any an;;lo =1 — sine of that angle. 
The chord of any angle = sine of A of that angle X 2. 

Example. = The ^^ide8 of a ri;^ht-anglod triangle are 3, 4 and 5 
feet ; re(|nired a side of the greatest square that may be cut from the 
triangle. 

A right line that bisects the right angle and extends to the hypot- 
enuse is equal to and becomes the diagonal of the required square, 

and the diagonal of any square multiplied by — -^ = a side of that 
square; then 5 : 1 :: 3 : .6 = sin 36"^ 51', and 

Siii'M)^4-30^5r — 45^ = .9S91) : 3 :: sin 00^ — 36^51'= .8 : 2.4245 
= .liagonal, and 2.4245 X .70711 =» 1.7144 = 8ide of wiuarc. Aris. 

Or sin 90" — 36*5r(.8) : 2.4245 : sin 45^(.70711) : 2.143, and 
(5 — 2J4lf _ 2.4245') -^ 4 X 2 = .2855 + ^ = ^.2855, and 



^(f) _ 2.143' — 2.2855") = 1.714 feet, side of square. Ans. 

Recapitulation. — AVhen the given angle is greater than 45®, 
its sine is expressed by tlie cosine of the angle which is the dilTer- 
ence oiDO^and the given angle : thus the sine of 40"^ is the tabular 
cosine of 5)0— 40 i= cosine of 44° : and the sine of G2° 50' is the cosine 
of 90—02^ 50' =z tabular cosine of 27^ 10', &c. 

To obtain the tangent of an angle that is greater than 45^, divide 
the tabular cosine of the angle which expresses the difference of 90° 
and the given angle by the tabular sine of that difference ; thus the 

. r .n. cos (90— 46)= cos 44® 71934 

tanirent of 46^== ^ --^tt^ =1^777^1^ = 1-^355, and 

sin 44'' 09466 

the tanizent of 

680 40' - ^o^ (!>0-'- 5 8^40 ) = cos 31° 2(y 85.m 

^" *" — sin 31° 20' 52002 ^•"'*- "'"^ 



TABLE OP SQUAKES, CUBES, SQUARE AND CUBE ROOTS. 245 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


1 


1 


1 ■ 


1.0000000 


1.0000000 


2 


4 


8 


1.4142136 


1.2599210 


3 


9 


27 


1.7320508 


1.4422496 


4 


16 


64 


2.0000000 


1.5874011 


5 


25 


125 


2.2360680 


1.7099759 


6 


36 


216 


2.4494897 


1.8171206 


7 


49 


343 


2.6457513 


1.9129312 


8 


64 


512 


2.8284271 


2.0000000 


9 


81 


729 


3.0000000 


2.0800837 


10 


100 


1000 


3.1622777 


2.1544347 


11 


121 


1331 


3.3166248 


2.2239801 


12 


144 


1728 


3.4641016 


2.2894286 


13 


169 


2197 


3.6055513 


2.3513347 


14 


196 


2744 


3.7416574 


2.4101422 


15 


225 


3375 


3.8729833 


2.4662121 


16 


256 


4096 


4.0000000 


2.5198421 


17 


289 


4913 


4.1231056 


2.5712816 


18 


324 


5832 


4.2426407 


2.6207414 


19 


361 


6859 


4.3588989 


2.6684016 


20 


400 


8000 


4.4721360 


2.7144177 


21 


441 


9261 


4.5825757 


2.7589243 


22 


484 


10648 


4.6904158 


2.8020393 


23 


529 


12167 


4.7958315 


2.8438670 


24 


576 


13824 


4.8989795 


2.8844991 


25 


625 


15625 


5.0000000 


2.9240177 


26 


676 


17576 


5.0990195 


2.9624960 


27 


729 


19683 


5.1961524 


3.0000000 


28 


784 


21952 


5.2915026 


3.0365889 


29 


841 


24389 


5.3851648 


3.0723168 


30 


900 


27000 


5.4772256 


3.1072325 


31 


961 


29791 


5.5677644 


3.1413806 


32 


1024 


32768 


5.6568542 


3.1748021 


33 


1089 


35937 


5.7445626 


3.2075343 


34 


1156 


39304 


5.8309519 


3.2396118 


35 


1225 


42875 


5.9160798 


3.2710663 


36 


1296 


46656 


6.0000000 


3.3019272 


37 


1369 


50653 


6.0827625 


3.3322218 


38 


1444 


54872 


6.1644140 


3.3619754 


39 


1521 


59319 


6.2449980 


3.3912114 


40 


1600 


64000 


6.3245553 


3.4199519 


41 


1681 


68921 


6.4031242 


3.4482172 


42 


1764 


74088 


6.4807407 


3.4760266 



21 * 



246 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



Number. 


Square. 


Cube. 


Square Boot. 


Cube Hoot. 


43 


1840 


79507 


6.5574385 


3.5033981 


44 


19:36 


85184 


6.6332496 


3.5303483 


45 


2025 


91125 


6.7082039 


3.5568933 


46 


2116 


97336 


6.7823300 


3.5830479 


47 


2209 


103823 


6.8556546 


3.6088261 


48 


2304 


110592 


6.9282032 


3.6342411 


49 


2401 


117649 


7.0000000 


3.6593057 


50 


2500 


125000 


7.0710678 


3.6840314 


51 


2601 


132651 


7.1414284 


3.7084298 


52 


2704 


140608 


7.2111026 


8.7325111 


53 


2809 


148877 


7.2801099 


3.7562858 


54 


2916 


157464 


7.3484692 


3.7797631 


55 


3025 


16<;375 


7.4161985 


3.8029525 


5G 


3136 


175616 


7.4833148 


8.8258624 


57 


3249 


185193 


7.5498344 


3.8485011 


58 


3364 


195112 


7.6157731 


8.8708766 


59 


3481 


205379 


7.6811457 


3.8929965 


60 


8600 


216000 


7.7459667 


3.9148676 


61 


3721 


226981 


7.8102497 


3.9304972 


62 


3844 


238328 


7.8740079 


3.9578915 


63 


3969 


250047 


7.9372539 


3.9790571 


64 


4096 


262144 


8.0000000 


4.0000000 


65 


4225 


271625 


8.0622577 


4.0207256 


66 


4356 


287496 


8.1240384 


4.0412401 


67 


4489 


300763 


8.1853528 


4.061.")1SO 


68 


4624 


314432 


8.2462113 


4.0816551 


69 


4761 


328509 


8.3066239 


4.1015661 


70 


4900 


343000 


8.3666003 


4.1212853 


71 


5041 


357911 


8.4261498 


4.1408178 


72 


5184 


373248 


8.4852814 


4.1601676 


73 


5329 


389017 


8.5440037 


4.1793390 


74 


5476 


405224 


8.6023253 


4.1983364 


75 


5625 


421875 


8.6602540 


4.2171633 


76 


5776 


438976 


8.7177079 


4.2358236 


77 


5929 


456533 


8.7749644 


4.2543210 


78 


6084 


474552 


8.8317609 


4.2726586 


79 


6241 


493039 


8.8881944 


4.2908404 


80 


6400 


512000 


8.9442719 


4.3088695 


81 


6561 


531441 


9.0000000 


4.3267487 


82 


6724 


551368 


9.0553851 


4.3444815 


83 


6889 


571787 


9.1104336 


4.3620707 


84 


7056 


592704 


9.1651514 


4.3795191 



TABLE OP SQUARES, CUBES, SQUARE AND CUBE ROOTS. 247 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


85 


7225 


614125 


9.2195445 


4.3968296 


86 


7396 


636056 


9.2736185 


4.4140049 


87 


7569 


658503 


9.3273791 


4.4310476 


88 


7744 


681472 


9.3808315 


4.4479602 


89 


7921 


704969 


9.4339811 


4.4647451 


90 


8100 


729000 


9.4868330 


4.4814047 


91 


8281 


753571 


9.5393920 


4.4979414 


92 


8464 


778688 


9.5916630 


4.5143574 


93 


8649 


80435 7 


9.6436508 


4.5306549 


94 


8836 


830584 


9.6953597 


4.5468359 


95 


9025 


857374 


9.7467943 


4.5629026 


96 


9216 


884736 


9.7979590 


4.5788570 


97 


9409 


912673 


9.84885 78 


4.5947009 


98 


9604 


941192 


9.8994949 


4.6104363 


99 


. 9S01 


970209 


9.0498744 


4.6260650 


100 


10000 


1000000 


10.0000000 


4.6415888 


101 


10201 


1030301 


10.0498756 


4.6570095 


102 


10404 


1061208 


10.0995049 


4.6723287 


103 


10609 


1092727 


10.1488916 


4.6875482 


104 


10816 


1124864- 


10.1980390 


4.7026694 


105 


11025 


1157625 


10.2469508 


4.7176940 


106 


11236 


1191016 


10.2956301 


4.7326235 


107 


11449 


1225043 


10.3440804 


4.7474594 


108 


11664 


1259712 


10.3023048 


4.7622032 


109 


11881 


1295029 


1074403065 


4.7768562 


110 


12100 


1331000 


10.4880885 


4.7914199 


111 


12321 


1367631 


10.5356538 


4.8058995 


112 


12544 


1404928 


10.5830052 


4.8202845 


113 


12769 


1442897 


10.6301458 


4.8345881 


114 


12996 


1481544 


10.6770783 


4.8488076 


115 


13225 


1520875 


10. 7238053 


4.8629442 


116 


13456 


15G0896 


10.7703296 


4.8760990 


117 


13689 


1601613 


10.8166538 


4.8909732 


118 


13924 


1643032 


10.8627805 


4.9048681 


119 


14161 


1685159 


10.9087121 


4.9186847 


120 


14400 


1728000 


10.9544512 


4.9324242 


121 


14641 


1771561 


11.0000000 


4.9460874 


122 


14884 


1815848 . 


11.0453610 


4.9596757 


123 


15129 


1860867 


11.0005365 


4.9731898 


124 


15376 


1906624 


11.1355287 


4.9866310 


125 


15625 


1953125 


11.1803399 


5.0000000 


126 


15876 


2000376 


11.2249722 


5.0132979 



248 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


127 


16129 


2048383 


11.2694277 


6.0265257 


128 


16384 


2097152 


11.3137085 


5.0396842 


129 


16641 


2146689 


11.3578167 


5.0527743 


130 


16900 


2197000 


11.4017543 


5.0657970 


131 


17161 


2248091 


11.4455231 


5.0787531 


132 


17424 


2299968 


11.4891253 


5.0916434 


133 


17689 


2352637 


11.5325626 


5.1044687 


134 


17956 


2406104 


11.5758369 


5.1172299 


135 


18225 


2460375 


11.6189500 


5.1299278 


130 


18496 


2515457 


11.6619038 


5.1425632 


13 7 


18769 


2571353 


11.7046999 


5.1551367 


138 


19044 


26280 72 


11.7473444 


5.1676493 


130 


19321 


2685619 


11.7898261 


5.1801015 


140 


19600 


2744000 


11.8321596 


5.1924941 


141 


19881 


2803221 


11.8743421 


5.2048279 


142 


20164 


2863288 


11.9163753 


5.2171034 


143 


20449 


2924207 


11.9582607 


5.2293215 


144 


20736 


2085984 


12.0000000 


6.2414828 


145 


21025 


3048625 


12.0415946 


6.2635879 


146 


21316 


3112136 


12.0830460 


6.2656374 


147 


21609 


3176523 


12.1243557 


6.2776321 


148 


21904 


3241792 


12.1655251 


6.2895725 


149 


22201 


3307949 


12.2065556 


6.3014592 


150 


22500 


33750U0 


12.2474487 


6.3132928 


151 


22801 


3442951 


12.2882057 


5.3250740 


152 


23104 


3511008 


1 2.3 28 S 280 


5.3368033 


153 


23409 


3581577 


12.3693169 


5.3484812 


154 


23716 


3652264 


12.4096736 


6.3601084 


155 


24025 


3723875 


12.4498996 


6.3716854 


156 


24336 


3796416 


12.4809960 


6.3832126 


157 


24649 


3869893 


12.5299641 


6.30460U7 


158 


24964 


3944312 


12.5698051 


6.4061202 


159 


25281 


4019679 


12.6095202 


6.4175015 


IGO 


25600 


4096000 


12.6491106 


6.4288352 


IGl 


25921 


4173281 


12.6885775 


6.4401218 


162 


26244 


4251528 


12.7270221 


5.4513618 


163 


26569 


4330747 


12.7671453 


5.4625556 


164 


26896 


4410944 


12.8062485 


5.4737037 


165 


27225 


4492125 


12.8452326 


6.4848066 


166 


27556 


4574296 


12.8840987 


6.4058647 


167 


27889 


4657463 


12.9228480 


5.5068784 


168 


28224 


4741632 


12.9614814 


6.5178484 



(TABLE OF SQtJAEES, CUBES, SQUARE AND CUBE ROOTS. 249 



Kumber. 


Square. 


Cube. 


Square Root. 


Cube Root. 


169 


28561 


4826809 


13.0000000 


5.5287748 


170 


28900 


4913000 


13.0384048 


5.5396583 


171 


29241 


5000211 


13.0766968 


5.5504991 


172 


29584 


5088448 


13.1148770 


5.5612978 


173 


29929 


5177717 


13.1529464 


5.5720546 


174 


30276 


5268024 


13.1909060 


5.5827702 


175 


30625 


5359375 


13.2287566 


5.5934447 


176 


30976 


6451776 


> 13.2664992 


5.6040787 


177 


31329 


5545233 


13.3041347 


5.6146724 


178 


31684 


5639752 


13.3416641 


5.6252263 


179 


32041 


5735339 


13.3790882 


5.6357408 


180 


32400 


5832000 


13.4164079 


5.6462162 


181 


32761 


5929741 


13.4536240 


5.6566528 


182 


33124 


6028568 


13.4907376 • 


5.6670511 


183 


33489 


6128487 


13.5277493 


5.6774114 


184 


33856 


6229504 


13.5646600 


5.6877340 


185 


34225 


•6331625 


13.6014705 


5.6980192 


186 


34596 


6434856 


13.6381817 


5.7082675 


187 


34969 


6539203 


13.6747943 


5.7184791 


188 


35344 


6644672 


13.7113092 


5.7286543 


189 


35721 


6751269 


13.7477271 


5.7387936 


190 


36100 


6859000 


13.7840488 


5.7488971 


191 


36481 


6967871 


13.8202750 


5.7589652 


192 


36864 


7077888 


13.8564065 


5.7689982 


193 


37249 


7189057 


13.8924400 


5.7789966 


194 


37636 


7301384 


13.9283883 


5.7889604 


195 


38025 


7414875 


13.9642400 


5.7988900 


196 


38416 


7529536 


14.0000000 


5.8087857 


197 


38809 


7645373 


14.0356688 


5.8186479 


198 


39204 


7762392 


14.0712473 


5.828486 7 


199 


39601 


7880599 


14.1067360 


5.8382725 


200 


40000 


8000000 


14.1421356 


5.8480355 


201 


40401 


8120601 


14.1774469 


5.8577660 


202 


40804 


8242408 


14.2126704 


5-8674673 


203 


41209 


8365427 


14.2478068 


5.8771307 


204 


41616 


8489664 


14.2828569 


5.8867653 


205 


42025 


8615125 


14.3178211 


5.8963685 


206 


42436 


8741816 


14.3527001 


5.9059406 


207 


42849 


8869743 


14.3874946 


5.9154817 


208 


43264 


8998912 


14.4222051 


5.9249921 


209 


43681 


9129329 


14.4568323 


5.9344721 


210 


44100 


9261000 


14.4913767 


5.9439220 



250 



TABLE OP SQUARES, CDBES, SQDAKB AN© CCBE ROOtS. 



Kumber. 


Square. 


Cube. 


Square Root. 


Cube Root. 


211 


44521 


9393931 


14.5258390 


5.9533418 


212 


44944 


9528128 


14.5602198 


5.9627320 


213 


45369 


9663597 


14.5945195 


5.9720926 


214 


45796 


9800344 


14.6287388 


5.9814241 


215 


46225 


9938375 


14.6628783 


5.9907264 


216 


46656 


10077696 


14.69G9385 


6.0000000 


217 


4 7089 


10218313 


14.7309199 


6.0092450 


218 


47524 


10360232 


14.7648231 


6.0184617 


219 


47961 


10503459 


14.7986486 


6.0276502 


220 


48400 


10648000 


14.8323970 


6.0368107 


221 


48841 


10793861 


14.8660687 


6.0459435 


222 


49284 


10941048 


14.8996644 


6.0550489 


223 


49729 


11089567 


14.9331845 


6.0641270 


224 


50176 


11239424 


14.9666295 


6.0731779 


225 


50625 


11390625 


15.0000000 


6 0822020 


22G 


51076 


11543176 


15 0332964 


6.0911994 


227 


51529 


11697083 


15.0665192 


6.1001702 


228 


51984 


11852352 


15.0996689 


6.1091147 


229 


52441 


12008989 


15.1327460 


6.1180332 


230 


52900 


12167000 


15.1657509 


6.1269257 


231 


53361 


12326391 


15.1986842 


6.1357924 


232 


53824 


12487168 


15.2315462 


6.1446337 


233 


54289 


12649337 


15.2643375 


6.15,34495 


234 


54756 


1281 2904 


15.2970585 


6.1622401 


235 


55225 


12977875 


15.3297097 


6.1710058 


23G 


55696 


13144256 


15.3622915 


6.1797466 


237 


56169 


13312053 


15.3948043 


6.1884628 


238 


56644 


13481272 


15.4272486 


6.1971544 


239 


57121 


13651919 


15.4596248 


6.2058218 


240 


57600 


13824000 


15.4919334 


6.2144650 


241 


58081 


13997521 


15.524174 7 


6.2230843 


242 


58564 


14172-188 


15.5563492 


6.2316797 


243 


59049 


14348907 


15.5884573 


6.2402515 


244 


59536 


14526784 


15.6204994 


6.2487998 


245 


60025 


14706125 


15.6524758 


6.2573248 


246 


60516 


14886936 


15.6843871 


6.26.^8266 . 


247 


61009 


15069223 


15.7162336 


6.2743054 


248 


61504 


15252992 


15.7480157 


6.2827613 


249 


62001 


15438249 


15.7797338 


6.2911.046 


250 


62500 


15625000 


15.8113883 


6.2996053 


251 


63001 


15813251 


15.8429795 


6.3079935 


252 


63504 


16003008 


15.8745079 


6.3163596 



i 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 251 



Number. 


Square. 


Cube. 


Square Root. 


Cube Ptoot. 


253 


64009 


16194277 


15.9059737 


6.3247035 


254 


64516 


16387064 


15.9373775 


6.3330256 


255 


65025 


16581375 


15.9687194 


6.3413257 


25G 


65536 


16777216 


16.0000000 


6.3496042 


257 


66049 


16974593 


16.0312195 


6.3578611 


258 


66564 


17173512 


16.0623784 


6.3660968 


259 


67081 


17373979 


16.0934769 


6.3743111* 


260 


67600 


17576000 


16.1245155 


6.3825043 


261 


68121 


17779581 


16.1554944 


6.3906765 


262 


68644 


17984728 


16.1864141 


6.3988279 


263 


69169 


18191447 


16.2172747 


6.4069585 


264 


69696 


18399744 


16.2480768 


6.4150687 


265 


70225 


18609625 


16.2788206 


^6.4231583 


266 


70756 


18821096 


16.3095064 


6.4312276 


267 


71289 


19034163 


16.3401346 


6.4392767 


268 


71824 


19248832 


16.3707055 


6.4473057 


269 


72361 


19465109 


16.4012195 


6.4553148 


270 


72900 


19683000 


16.4316767 


6.4633041 


271 


73441 


19902511 


16.4620776 


6.4712736 


272 


73984 


20123648 


16.4924225 


6.4792236 


273 


74529 


20346417 


16.5227116 


6.4871541 


274 


75076 


205 70824 


16.5529454 


6.4950653 


275 


75625 


20796875 


16.5831240 


6.5029572 


276 


76176 


21024576 


16.6132477 


6.5108300 


277 


76729 


21253933 


16.6433170 


6.5186839 


278 


77284 


21484952 


16.6783320 


6.5265189 


279 


77841 


21717639 


16.7032931 


6.5343351 


280 


78400 


21952000 


16.7332005 


6.5421326 


281 


78961 


22188041 


16.7630546 


6.5499116 


282 


79524 


22425768 


16.7928556 


6.5576722 


283 


80089 


22665187. 


16.8226038 


6.5654144 


284 


80656 


22906304 


16.8522995 


6.5731385 


285 


81225 


23149125 


16.8819430 


6.5808443 


286 


81796 


23393656 


16.9115345 


6.5885323 


287 


82369 


23639903 


16.9410743 


6.5962023 


288 


82944 


23887872 


16.9705627 


6.6038545 


289 


83521 


24137569 


17.0000000 


6.6114890 


290 


84100 


24389000 


17.0293864 


6.6191060 


291 


84681 


24642171 


17.0587221 


6.6267054 


292 


85264 


24897088 


17.0880075 


6.6342874 


293 


85849 


25153757 


17.1172428 


6.6418522 


294 


86436 


25412184 


17.1464282 


6.6493998 



252 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


295 


87025 


25672375 


17.1755640 


6.6569302 


29G 


87G16 


25934336 


17.2046505 


^6.6644437 


297 


88209 


26198073 


17.2336879 


6.6719403 


298 


88804 


26463592 


17.2626765 


6.6 794200 


299 


89401 


26730899 


17.2916165 


6.6868831 


300 


90000 


27000000 


17.3205081 


6.6943295 


301 


90601 


27270901 


17.3493516 


6.7017593 


302 


91204 


27543608 


17.3781472 


6.7091729 


303 


91809 


27818127 


17.4068955S 


6.7165700 


304 


92416 


28094464 


17.4355958 


6.7239508 


305 


93025 


28372625 


17.4642492 


6.7313155 


306 


93636 


28652616 


17.4928557 


6.7386643 


30 7 


*4249 


28934443 


17.5214155 


6.7459967 


308 


94864 


29218112 


17.5499288 


6.7533134 


309 


95481 


29503629 


17.5783958 


6.7606143 


310 


96100 


29791000 


17.6068169 


6.7678995 


311 


96721 


30080231 


17.6351921 


6.7751690 


312 


97344 


30371328 


17.6635217 


6.7824 229 


313 


97969 


30664297 


17.6918060 


6.7896613 


314 


98596 


30959144 


17.7200451 


6.7968S44 


315 


99225 


31255875 


17.7482393 


6.8040921 


31G 


09856 


31554496 


17.7763888 


6.8112847 


317 


100489 


31.S55013 


17.8044938 


6.8184620 


318 


101124 


32157432 


17.8325546 


6.8256242 


319 


101761 


32461759 


17.8605711 


6.8327714 


320 


102400 


32768000 


•17.8885438 


6.8399037 


321 


103041 


33076161 


17.9164729 


6.8470213 


322 


103684 


83386248 


17.9443584 


6.8541 •40 


323 


104329 


33698267 


17.9722008 


6.8612120 


324 


104976 


34012224 


18.0000000 


6.8682855 


325 


105625 


3432gl25 


18.0277564 


6.8753433 


326 


106276 


34645976 


18.0554701 


6.882.>s,s8 


327 


106929 


34965783 


18.0831413 


6.8894188 


328 


107584 


352>^7552 


18.1107703 


6.8964345 


329 


108241 


35611289 


18.1383571 


6.9034359 


330 


108900 


35987000 


18.1659021 


6.9104232 


331 


109561 


36264691 


18.1934054 


6.9173964 


332 


110224 


36594368 


18.2208672 


6.9243556 


333 


110889 


36926037 


18.2482876 


6.9313088 


334 


111556 


37259704 


18.2756669 


6.9382321 


335 


112225 


37595375 


18.3030052 


6.9451496 


336 


112896 


37933056 


18.3303028 


6.9520533 



^TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 253 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


337 


113569 


38272753 


18.3575598 


6.9589434 


338 


114244 


38614472 


18.3847763 


6.9658198 


339 


114921 


38958219 


18.4119526 


6.9726826 


340 


115600 


39304000 


18.4390889 


6.9795321 


341 


116281 


39651821 


18.4661853 


6.9863681 


342 


116964 


40001688 


18.4932420 


6.9931906 


343 


117649 


40353607 


18.5202592 


7.0000000 


344 


118336 


40707584 


18.5472370 


7.0067962 


845 


119025 


41063625 


18.5741756 


7.0135791 


34G 


119716 


41421736 


18.6010752 


7.0203490 


347 


120409 


41781923 


18.6279360 


7.0271058 


348 


121104 


42144192 


18.6547581 


7.0338497 


349 


121801 


42508549 


18.6815417 


7.0405860 


350 


122500 


42875000 


18.7082869 


7.0472987 


351 


123201 


43243551 


18.7349940 


7.0540041 


352 


123904 


43614208 


18.7616630 


7.0606967 


353 


124609 


43986977 


18.7882942 


7.0673767 


354 


125316 


44361864 


18.8148877 


7.0740440 


355 


126025 


44738875 


18.8414437 


7.0806988 


356 


126736 


45118016 


18.8679623 


7.0873411 


357 


127449 


45499293 


18.8944436 


7.0939709 


358 


128164 


45882712 


18.9208879 


7.1005885 


359 


128881 


46268279 


18.9472953 


7.1071937 


3G0 


129600 


46656000 


18.9736660 


7.1137866 


361 


130321 


47045881 


19.0000000 


7.1203674 


362 


131044 


47437928 


19.0262976 


7.1269360 


363 


131769 


47832147 


19.0525589 


7.1334925 


364 


132496 


48228544 


19.0787840 


7.1400370 


365 


133225 


48627125 


19.1049732 


7.1465695 


366 


133956 


49027896 


19.1311265 


7.1530901 


367 


134689 


49430863 


19.1572441 


7.1595988 


368 


135424 


49836032 


19.1833261 


7.1660957 


369 


136*161 


50243409 


19.2093727 


7.1725809 


370 


136900 


50653000 


19.2353841 


7.1790544 


371 


137641 


51064811 


19.2613603 


7.1855162 


372 


188384 


51478848 


19.2873015 


7.1919663 


373 


139129 


51895117 


19.3132079 


7.198405O 


374 


139876 


52313624 


19.3390796 


7.2048322 


375 


140625 


52734375 


19.3649167 


7.2112478 


376 


141376 


53157376 


19.3907194 


7.2176522 


377 


142129 


53582633 


19.4164878 


7.2240450 


, 378 


142884 


54010152 


19.4422221 


7.2304268 

— a 



22 



254 TABLE OF SQUAHES, CUBES, SQUABE AND CUBE ROOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Boot. 


37a 


143G41 


54439939 


19.4679223 


7.2367972 


380 


144400 


64872000 


19.4935887 


7.243 15G5 


381 


145161 


55306341 


19.5192^13 


7.2495045 


382 


145924 


65742968 


19.5448203 


7.2558415 


383 


14CG89 


56181887 


19.5703858 


7.2621G75 


384 


147456 


66623104 


19.5959179 


7.2684824 


385 


148225 


67066625 


19.6214169 


7.274 78G4 


38G 


148996 


67512456 ' 


19.6468827 


7.2810794 


387 


1497C9 


67960603 


19.6723156 


7.2873G17 


388 


150544 


. 68411072 


19.6977156 


7.293(i3;{0 


389 


151321 


68863869 


19.7230829 


7.29989;J6 


390 


152100 


69319000 


19.7484177 


7.3061 4 :JG 


391 


152881 


697Z6471 


19.7737199 


7.3123S2S 


392 


153GG4 


60236288 


19.7989899 


7.3186114 


393 


154449 


60698457 


19.8242276 


7.3248295 


394 


J 55236 


61162984 


19.8494332 


7.331 0,3(;d 


395 


15G025 


61629875 


19.8746069 


7.3372339 


396 


15G816 


62099136 


19.8997487 


7.3434205 


397 


157609 


62570773 


19.9218588 


7.3495966 


398 


158404 


63044792 


19.9499378 


7.3557624 


399 


159201 


63521199 


19.9749844 


7.3619178 


400 


160000 


64000000 


20.0000000 


7.3680630 


401 


160801 


64481201 


20.0249844 


7.3741979 


402 


161604 


64964808 


20.0499377 


7.3.S03227 


403 


162409 


65450827 


20.0748599 


7.38G4373 


404 


163216 


65939264 


20.0997512 


7.31)25418 


405 


164025 


66430125 


20.1246118 


7.3986363 


406 


1G4836 


66928416 


20.1494417 


7.404 7206 


407 


165649 


67419143 


20.1742410 


7.4107950 


, 408 


166464 


67917312 


20.1990099 


7.4168595 


409 


167281 


68417929 


20.223-7484 


7.4229142 


410 


168100 


68921000 


20.2484567 


7.4289589 


411 


168921 


69426531 


20.2731319 


7.4349938 


412 


169744 


69934528 


20.297 7831 


7.4410189 


413 


170569 


70444997 


20.3224014 


7.4470^43 


414 


171396 


70957944 


20.3469899 


7.4530399 


415 


172225 


71473375 


20.3715488 


7.4590359 


41G 


173056 


71991296 


20.3960781 


7.4G50223 


417 


173889 


72511713 


20.4205779 


7.4709991 


418 


174724 


73034632 


20.4450483 


7.47696G4 


419 


1755G1 


735G0059 


20.4694895 


7.4829242 


420 


17G400 


74088000 


20.4939015 


7.4888724 



TABLE OF SQUAIIIDS, C0BE3, SQUARi: AND CUBE ROOTS. 255 



Number. 


Square. 


Cube. 


Square Root. 


Cube Hoot. 


421 


177241 


74618461 


20.5182845 


7.4948113 


422 


1 78084 


75151448 


20.5426336 


7.5007406 


423 


1 78929 


7568696 7 


20.5669638 


7.5066607 


424 


179776 


76225021 


20.5912603 


7.5125715 


425 


180625 


76 765625. 


20.6155281 


7.5184730 


426 


181476 


77308776 


20.6397674 


7.5243652 


427 


182329 


77854483 


20.6639 783 


7.5302482 


428 


1 83 1 84 


78402 752 


20.6881609 


7.5361221 


429 


184041 


78953589 


20.7123152 


7.5419867 


430 


181900 


70507000 


20.73G4414 


7. .H 78423 


431 


185761 


80062991 


20.7605395 


7.5536888 


432 


186621 


80621568 


20.7846097 


7.5595263 


433 


187489 


81182737 


21J.8086520 


7.5653548 


434 


188356 


81746504 


20.8326667 


7.5711743 


435 


189225 


82312875 


20.8566536 


7.5769849 


43G 


190096 


82881856 


20.8806130 


7.5827865 


437 


190969 


83453453 


20.9045450 


7.5885793 


438 


191844 


840276 72 


20.92o4'495 


7.5943633 


439 


192721 


84604519 


20.9523268 


7.6001385 


440 


193600 


85181000 


20.9761770 


7.6059049 


441 


194481 


85766121 


21.0000000' 


7.6116626 


442 


195364 


8635(;888 


21.0237960 


7.6174116 


443 


196249 


86938307 


21.0475652 


7.6231519 


444 


197136 


87528384 


21.0713075 


7.6288837 


445 


198025 


88121125 


21.0950231 


7.6346067 


446 


198916 


88716536 


21.1187121 


7.6403213 


447 


199809 


89314623 


21.1423745 i 


7.6460272 


448 


200704 


899153i)2 


21.1660105 


7.6517247 


449 


201601 


90518849 


21.1896201 


7.6574138 


450 


202500 


91125000 


21.2132034 


7.6630943 


451 


203401 


91733851 


21.2367606 


7.6687665 


452 


204304 


92345408 


21.2602916 


7.6744303 


453 


205209 


929596 7 7 


21.2.^37967 


7.6800857 


454 


206116 


935 76664 


21.3072758 


7.6857328 


455 


207025 


94196375 


21.3307290 ! 


7.6913717 


456 


207936 


94818816 


21.3541565 


7.6970023 


457 


208849 


95443993 


21.3775583 


7.7026246 


458 


209764 


96071912 


21.4009346 


7.7082388 


459 


210681 


96702579 


21.4242853 


7.7138448 


460 


211600 


97336000 


21.4476106 


7.7194426 


461 


212521 


97972181 


21.4709106 


7.7250325 


462 


213444 


98611128 


21.4941853 


7.7306141 



256 TABLE or SQUARES, CUBES, SQUARE AND CUBE BOOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


463 


214369 


99252847 


21.5174348 


7.7361877 


464 


215296 


99897344 , 


21.5406592 


7.7417532 


465 


216225 


100544625 


21.5638587 


7.7473109 


466 


217156 


101194696 


21.5870331 


7.7528606 


467 


2180K9 


101847563 


21.6101828 


7.7584023 


468 


219024 


1025()3232 


21.6333077 


7.7639361 


460 


219961 


H)3161709 


21.6564078 


7.7694620 


470 


220900 


103H23000 


21.6794834 


7.7749801 


471 


221841 


10I4S7111 


21.7025344 


7.7804904 


472 


222784 


10:>154(M8 


21.7255610 


7.7859928 


473 


223729 


10,^S23817 


21.7485632 


7.7914875 


474 


2246 76 1 


106496424 


21.7715411 


7.7969745 ♦ 


475 


225625 


107171H75 


21.7944947 


7.8024538 


476 


226576 


107850176 


21.8174242 


7.8079254 


477 


227529 


108r)31333 


21.8403297 


7.8133892 


478 


228484 


109215352 


21.8632111 


7.8188456 


479 


229441 


109902239 


21. 8H606H6 


7.S242942 


48P 


230400 


11 Of) 02000 


21.9089023 


7.8297358 


481 


231361 


111284641 


21.9317122 


7.8351688 


482 


232324 


1119S0168 


21.9544984 


7.8405949 


483 


233289 


112678587 


21.9772610 


7.8460134 


484 


234256 


113379904 


22.0000000 


7.8514244 


485 


235225 


114084125 


22.0227155 


7.8568281 


486 


236196 


114 791256 


22.0454077 


7.8622242 


487 


237169 


115501303 


22.0680765 


7.8676130 


488 


238144 


116214272 


22.0907220 


7.8729944 


489 


239121 


116930169 


22.1133444 


7.8783684 


490 


240100 


117649000 


22.1359436 


7.8837352 


491 


241081 


118370771 


22.1585198 


7.889094a 


492 


242064 


119095488 


22.1810730 


7.8944468 


493 


243049 


119823157 


22.2036033 


7.8997917 


494 


244036 


120553784 


22.2261108 


7.9051294 


495 


245025 


121287375 


22.2485955 


7.9104599 


496 


246016 


122023936 


22.2710575 


7.9157832 


497 


247009 


122763473 


22.2934968 


7.9210994 


498 


248004 


123505992 


22.3159136 


7.9264085 


499 


249001 


124251499 


22.3383079 


7.9317104 


500 


250000 


125000000 


22.3606798 


7.9370053 


501 


251001 


125751501 


22.3830293 


7.9422931 


502 


252004 


126506008 


22.4053565 


7.9475739 


503 


253009 


127263527 


22.4276615 


7.95284 77 


504 


254016 


128024064 


22.4499448 


7.9581144 



TABLE m SQUARES, CUBES, SQUARE AND CUBE ROOTS. 257 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


505 


255025 


128787625 


22.4722051 


7.9633743 


506. 


25G036 


129554216 


22.4944438 


7.9686271 


507 


257049 


130323843 


22.5166605 


.7.9738731 


508 


258064 


131096512 


22.5388553 


7.9791122 


509 


259081 


131872229 


22.5610283 


7.9843444 


510 


260100 


132051000 


22.5831796 


7.9895697 


511 


261121 


133432831 


22.6053091 


7.9947883 


512 


262144 


134217728 


22.6274170 


8.0000000 


513 


263169 


135005697 


22.6495033 


8.0052049 


514 


264196 


135796744 


22.6715681 


8.0104032 


515 


265225 


136590875 


22.6936114 


8.0155946 


516 


266256 


137388096 


22.7156334 


8.0207794 


517 


267289 


138188413 


22.7376341 


8.0259574 


518 


268324 


138991832 


22.7596134 


8.0311287 


519 


269361 


139798359 


22.7815715 


8.0362935 


520 


270400 


140608000 


22.8035085. 


8.0414515 


521 


271441 


141420761 


22.8254244 


8.0466030 


522 


272484 


142236648 


22.8473193 


8.0517479 


523 


273529 


143055667 


22.8691933 


8.0568862 


524 


274576 


143877824 


22.8910463 


8.0620180 


525 


275625 


144703125 


22.9128785 


8.0671432 


526 


276676 


145531576 


22.9346899 


8.0722620 


527 


277729 


146363183 


22.9564806 


8.0773743 


528 


278784 


147197952 


22.9782506 


8.0824800 


529 


279841 


148035889 


23.0000000 


8.0875794 


530 


280900 


148877000 


23.0217289 


8.0926723 


531 


281961 


149721291 


23.0434372 


8.0977589 


532 


2.83024 


150568768 


23.0651252 


8.1028390 


533 


284089 


151419437 


23.0867928 


8.1079128 


534 


285156 


152273304 


23.1084400 


8.1129803 


535 


286225 


153130375 


23.1300670 


8.1180414 


536 


287296 


153990656 


23.1516738 


8.1230962 


537 


288369 


154854153 


23.1732605 


8.1281447 


538 


289444 


155720872 


23.1948270 


8.1331870 


539 


290521 


156590819 


23:2163735 


8.1382230 


540 


291600 


157464000 


23.23 79001 


8.1432529 


541 


292681 


158340421 


23.2594067 


8.1482765 


542 


293764 


159220088 


23.2808935 


8.1532939 


543 


294849 


160103007 


23.3023604 


8.1583051 


544 


295936 


160989184 


23.3238076 


8.1633102 


545 


297025 


161878625 


23.3452351 


8.1683092 


546 


298116 


162771336 


23.3666429 


8.1733020 



22* 



258 TABLE OF SQUARES, CUBES, SQUARE AND CUBF ROOTS. 



Number. ' 


Square. 


Cube. 


Square Itoot. 


Cube Boot. 


54 7 


299209 


163667325 


23.3880311 


8.1782888 


548 


300304 


164566592 


23.4093998 


8.1832695 


549 


301401 


165469149 


23.4307490 


8.1882441 


550 


302500 


166375000 


23.4520788 


8.1932127 


551 


303601 


IC 7284 151 


23.4 733892 


8.1981753 


552 


304704 


168196608 


23.4946802 


8.2031819 


553 


305809 


169112377 


23.5159520 


8.2080825 


554 


306916 


170031464 


23.5372046 


8.2130271 


555 


308025 


170953875 


23.5584380 


8.2179657 


556 


309136 


171879616 


23.5796522 


8.2228985 


557 


310249 


172808693 


23.6008474 


8.2278254 


558 


311364 


173741112 


23.6220236 


8.2327463 


559 


312481 


174676S79 


23.6431808 


8.2376614 


5G0 


313600 


17.-)6 16000 


23.6643191 


8.2425706 


561 


314721 


176558481 


23.6854886 


8.24 74 740 


562 


315844 


177504328 


23.7065892 


8.2523715 


563 


316969 


178453547 


23.7276210 


8.2572635 


564 


318096 


179406144 


23.7486842 


8.2621492 


665 


319225 


180362125 


23.7697286 


8.26 70294 


566 


320356 


181321496 


23.7907545 


8.2719039 


567 


321489 


182284263 


23.8117618 


8.2767726 


568 


322624 


183250432 


23.8327506 


8.2816255 


569 


323761 


184220009 


23.8537209 


8.2864928 


570 


324900 


185193000 


28.8746728 


8.2913444 


571 


326041 


186169411 


23.8956068 


8.2961903 


572 


327184 


187149248 


23.9165215 


8.3010304 


573 


328329 


188132517 


23.9374184 


8.3058651 


574 


329476 


189119224 


23.9582971 


8.8106941 


575 


830625 


190109375 


23.9791576 


^.3155175 


576 


331776 


191102976 


24.0000000 


8.8203353 


577 


332929 


192100033 


24.0208243^ 


8.32514 75 


578 


334084 


193100552 


24.0416306* 


8.3299542 


579 


335241 


194104539 


24.0624188 


8.334 7558 


580 


336400 


195112000 


24.0831891 


8.3395509 


581 


337561 


' 196122941 


24.1039416 


8.3443410 


582 


338724 


197137368 


22.1246762 


8.3491256 


583 


339889 


198155287 


24.1453929 


8.353904 7 


584 


341056 


199176704 


24.1660919 


8.3586784 


585 


342225 


200201625 


24.1867732 


8.3634466 


586 


343396* 


201230056 


24.2074369 


8.3682095 


587 


344569 


2O2262003 


24.2280829 


8.8729668 


588 


345744 


203297472 


24.2487113 


8.8777188 



II 



STABLE 01* SQUARES, CUBES, SQUARE AND CUBE ROOTS. 259 



Number. 


Square. 


Cube. 


Square Root. 


Cube Koot. 


589 


346921 


204336469 


24.2693222 


8.3824653 


590 


348100 


205379000 


24.2899156 


8.3872065 


591 


349281 


206425-071 


24.3104996 


8.3919428 


592 


350464 


207474688 


24.3310501 


8.3966729 


593 


351649 


208527857 


24.3515913 


8.4013981 


594 


352836 


209584584 


24.3721152 


8.4061180 


595 


354025 


210G44875 


24.3926218 


8.4108326 


596 


355216 


211708736 


24.4131112 


8.4155419 


597 


35G409 


212776173 


24.4335834 


8.4202460 


598 


357604 


213847192 


24.4540385 


8.4249448 


599 


358801 


214921799 


24.4744765 


8.4296383 


600 


3G0000 


216000000 


24.49489 74 


8.4343267 


601 


361201 


217081801 


24.5153013 


8.4390098 


602 


362404 


218167208 


24.5356883 


8.4436877 


603 


3G3G09 


219256227 


24.5560583 


8.4483605 


604 


364816 


220348864 


24.5764115 


8.4530280 


605 


3GG025 


221445125 


24.5967478 


8.4576906 


606 


3G7236 


222545016 


24.6170673 


8.4623479 


607 


368449 


223648543 


24.6373700 


8.4670001 


608 


369664 


224755712 


24.6576560 


8.4716471 


609 


370881 


225866529 


24.6779254 


8.4762892 


610 


372100 


226981000 


24.6981781 


8.4809261 


611 


373321 


228099131 


24.7184142 


8.4855579 


612 


374544 


229220928 


24.7386338 


8.4901848 


613 


375769 


230346397 


24.7588368 


8.4948065 


614 


376996 


231475544 


24.7790234 


8.4994233 


615 


378225 


232608375 


24.7991935 


8.5040350 


616 


379456 


233744896 


24.8193473 


8.5086417 


617 


380689 


234885113 


24.83i)4847 


8.5132435 


618 


381924 


23G029032 


24.8596058 


8.5178403 


619 


383161 


237176659 


24.8797106 


8.5224331 


620 


384400 


238328000 


24.8997992 


8.5270189 


621 


385641 


239483061 


24.9198716 


8.5316009 


622 


386884 


240641848 


24.9399278 


8.5361780 


623 


388129 


24180436 7 


24.9599679 


8.5407501 


624 


389376 


242970624 


24.9799920 


8.5453173 


625 


390625 


244140625 


25.0000000 


8.5498797 


626 


391876 


245314376 


25.0199920 


8.5544372 


627 


393129 


246491883 


25.0399681 


8.5589899 


628 


394384 


247673152 


25.0599282 


8.5635377 


629 


395641 


248858189 


25.0798724 


8.5680807 


630 


396900 


250047000 


25.0998008 


8.5726189 



2G0 TABLE OP SQUAHES, cubes, SQU-UIE AND CUBE BOOTS. 



Number. . 


Square. 


Cube. 


Square Root. 


« — 

Cube KoQt. 


631 


398161 


251239591 


25.1197134 


8.5771523 


632 


309424 


252435968 


25.1306102 


8.5816809 


633 


400689 


253636137 


25.1594913 


8.5862047 


634 


401956 


254840104 


25.1703566 


8.5907238 


635 


403225 


25604 7875 


25.1992063 


8.5952380. 


636 


404496 


257259456 


25.2190404 


8.599.7476 


637 


405769 


2584 74853 


25.2388589 


8.6042625 


638 


407044 


259694072 


25.2586619 


8.6087526 


639 


408321 


260917119 


25.2784493 


8.6132480 


640 


409600 


262141000 


25.2982213 


8.6177388 


641 


410881 


2633 74 721 


25.3179778 


8.6222248 


642 


412164 


264609288 


25.3377189 


8.626 7063 


643 


413419 


26584 7707 


25.3574447 


8.6311830 


644 


414736 


267089984 


25.3771551 


8.6356551 


645 


416025 


268336125 


25.3968502 


8.6401226 


646 


417316 


2695S6136 


25.4165302 


8.6445855 


647 


418609 


270840023 


25.4361947 


8.64 90437 


648 


419904 


272097792 


25.4558441 


8.6534974 


649 


421201 


273359449 


25.4 754 784 


8.65 79465 


650 


422500 


274625000 


25.4950976 


8.6623911 


651 


423801 


275894451 


25.5147013 


8.6668310 


652 


425104 


2771678U.S 


25.5342907 


8.6712665 


653 


426409 


278445077 


25.5538647 


8.6756974 


654 


427716 


279726264 


25.6734237 


8.6801237 


655 


429025 


281011375 


25.5929678 


8.6,S45456 


656 


' 430336 


282300416 


25.6124969 


8.6S.S9630 


657 


431649 


283593393 


25.6320112 


8.6933759 


658 


432964 


284890312 


25.6515107 


8.6977843 


659 


434281 . 


286191179 


25.6709953 


8.7021882 


660 


435600 


287496000 


25.6904652 


8.70G5877 


661 


436921 


288804 781 


25.7099203 


8.7109827 


662 


438244 


290117528 


25.7293607 


8.7153734 


663 


439569 


291434247 


25.7487864 


8.7197596 


664 


440896 


292754944 


25.7681075 


8.7241414 


665 


442225 


294079625 


25.7875939 


8.7285187 


666 


443556 


295408296 


25.8069758 


8.7328918 


667 


444889 


296740963 


25.8263431 


8.7372604 


668 


446224 


298077632 


25.8456060 


8.7416246 


GG9 


447561 


299418309 


25.8650343 


8.7450846 


G70 


448900 


300763000 


25.8843582 


8.7503401 


671 


450241 


302111711 


25.90366 77 


8.7546013 


672 


451584 


303464448 


25.9229628 


8.7590383 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE HOOTS. 261 



Number. 


Square. 


Cube. 


Square Koot. 


Cube Koot. 


673 


452929 


304821217 


25.9422435 


8.7633809 


674 


454276 


306182024 


25.9615100 


8.7677192 


675 


455625 


307546875 


25.9807621 


8.7720532 


676 


456976 


308915776 


26.0000000 


8.7763830 


677 


458329 


310288733 


26.0192237 


8.7807084 


678 


459684 


311665752 


26.0384331 


8.7850296 


679 


461041 


313046839 


26.0576284 


8.7893466 


680 


462400 


314432000 


26.0768096 


8.7936593 


681 


463761 


'315821241 


26.0959767 


8.7979679 


682 


465124 


317214568 


26.1151297 


8.8022721 


683 


466489 


318611987 


26.1342687 


8.8065 722 


684 


467856 


320013504 


26.1533937 


8.8108681 


685 


469225 


321419125 


26.1725047 


8.8151598 


686 


470596 


322828856 


26.1916017 


8.8194474 


687 


471969 


324242703 


26.2106848 


8.8237307 


688 


473344 


325660672 


26.2297541 


8.8280099 


689 


474721 


327082769 


26.2488095 


8.8322850 


690 


476100 


328509000 


26.2678511 


8.8365559 


691 


477481 


329939371 - 


26.2868789" 


8.8408227 


692 


478864 


331373888 


26.3058929 


8.8450854 


693 


480249 


332812557 


26.3248932 


8.8493440 


694 


481636 


334255384 


26.3438797 


8.8535985 


695 


483025 


335702375 


26.3628527 


8.8578489 


696 


484416 


337153536 


26.3818119 


8.8620952 


697 


485809 


338608873 


26.4007576 


8.8663375 


698 


487204 


340068392 


26.4196896 


8.8705757 


699 


488601 


341532099 


26.4386081 


8.8748099 


700 


490000 


343000000 


26.4575131 


8.8790400 


701 


491401 


344472101 


26.4764046 


8.8832661 


702 


492804 


345948408 


26.4^952826 


8.8874882 


703 


494209 


347428927 


26.5141472 


8.8917063 


704 


495616 


348913G64 


26.5329983 


8.8959204 


705 


497025 


350402625 


26.5518361 


8.9001304 


706 


498436 


351895816 


26.5'706605 


8.9043366 


707 


499849 


353393243 


26.5894716 


8.9085387 


708 


501264 


354894912 


26.6082694 


8.9127369 


709 


502681 


356400829 


26.6270539 


8.9169311 


710 


504100 


357911000 


26.6458252 


8.9211214 


711 


505521 


359425431 


26.6645833 


8.9253078 


712 


506944 


360944128 


26.6833281 


8,9294902 


713 


508369 


362467097 


26.7020598 


8.9336687 


714 


509796 


3639943t4 


26.7207784 


8.9378433 



262 TABLE OF SQUABES*, CUBES, SQUABB AND CUBE BOOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Boot. 


715 


511225 


365525875 


26.7394839 


8.9420140 


71G 


512G5G 


367061696 


26.7581763 


8.9461809 


717 


514089 


368601813 


26.7768557 


8.9503438 


718 


515524 


370146232 


26.7955220 


8.9545029 


719 


5l696r 


371694959 


26.8141754 


8.9586581 


720 


518400 


373248000 


26.8328157 


8.9628095 


721 


519841 


374805361 


26.8514432 


8.9669570 


722 


521284 


376367048 


26.8700577 


8.9711007 


723 


522729 


377933067 


26.8886593 


8.9752406 


724 


524176 


379503424 


26.9072481 


8.9793766 


725 


525625 


381078125 


26.9258240 


8.9835089 


72G 


527076 


382657176 


26.9443872 


8.9876373 


727 


528529 


384240583 


26.9629375 


8.9917620 


728 


529984 


385828352 


26.9814751 


8.9958899 


729 


531441 


387420489 


27.0000000 


9.0000000 


730 


532900 


389017000 


27.0185122 


9.0041134 


731 


534361 


390617891 


27.0370117 


9.0082229 


732 


535S24 


392223168 


27.0554985 


9.0123288 


733 


5372S9 


393832837 


27.0739727 


9.0164309 


734 


538756 


395446904 


27.0924344 


9.0205293 


735 


540225 


397065375 


27.1108834 


9.0246239 


736 


541696 


398688256 


27.1293199 


9.0287149 


737 


543169 


400315553 


27.1477149 


9.0328021 


738 


544644 


401947272 


27.1661554 


9.0368857 


739 


546121 


403583419 


27.1845544 


9.0409655 


740 


54 7600 


405224000 


27.2029140 


9.0450419 


741 


549081 


406869021 


27.2213152 


9.0491142 


742 


550564 


408518488 


27.2396769 


9.0531831 


743 


552049 


410172407 


27.2580263 


9.05724S2 


744 


553536 


411830784 


27.2763634 


9.0613098 


745 


555025 


413493625 


27.2946881 


9.0653677 


746 


556516 


415160936 


27.3130006 


9.0694220 


747 


558009 


416832723 


27.3313007 


9.0734726 


748 


559504 


418508992 


27.3495887 


9.0775197 


749 


561001 


420189749 


27.3678644 


9.0815631 


750 


562500 


421875000 


27.3861279 


9.0856030 


751 


564001 


423564751 


27.4043792 


9.0896392 


752 


565504 


425259008 


27.4226184 


9.0936719 


753 


567009 


426957777 


27.4408455 


9.0977010 


754 . 


568516 


4286G1064 


27.4590604 


9.1017265 


755 


570025 


430368875 


27.4772633 


9.1057485 


756 


571536 


432081^16 


27.4954542 


9.1097669 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 263 



Number. 


Square. 


Cube. 


/ 

Square Koot. 


Cube Reot. 


757 


573049 


433798093 


27.5136330 


9.1137818 


758 


574564 


435519512 


27.5317998 


9.1177931 


759 


576081 


437245479 


27.5499546 


9.1218010 


760 


577600 


438976000 


27.5680975 


9.1258053 


761 


579121 


440711081 


27.5862284 


9.1298061 


762 


580644 


442450728 


27.6043475 


9.1338034 


763 


582169 


444194947 


27.6224546 


9.1377971 


764 


583696 


445943744 


27.6405499 


9.1417875 


765 


585225 


447697125 


27.6586334 


9.1457742 


766 


586756 


449455096 


27.6767050 


9.1497576 


767 


588289 


451217663 


27.6947648 


9.1537375 


768 


589824 


452984832 


27.7128129 


9.1577139 


769 


591361 


454756609 


27.7308492 


9.1616869 


770 


592900 


456533000 


27.7488739 


9.1656565 


771 


594441 


458314011 


27.7668868 


9.1696225 


772 


595984 


460099648 


27.7848880 


9.1735852 


773 


597529 


461889917 


27.8028775 


9.1775445 


774 


599076 


463684824 


27.8208555 


9.1815003 


775 


600625 


465484375 


27.8388218 


9.1854527 


776 


602176 


467288576 


27.8567766 


9.1894018 


777 


603729 


469097433 


27.8747197 


9.1933474 


778 


605284 


470910952 


27.8926514 


9.1972897 


779 


606841 


472729139 


27.9105715 


9,2012286 


780 


608400 


474552000 


27.9284801 


9.2051641 


781 


609961 


476379541 


27.9463772 


9.2090962 


782 


611524 


478211768 


27.9642629 


9.2130250 


783 


613089 


480048687 


27.9821372 


9.2169505 


784 


614656 


481890304 


28.0000000 


9.2208726 


785 


616225 


483736625 


28.0178515 


9.2247914 


786 


617796 


485587656 


28.0356915 


9.2287068 


787 


619369 


487443403 


28.0535203 


9.2326189 


788 


620944 


489303872 


28.0713377 


9.2365277 


789 


622521 


491169069 


28.0891438 


9.2404333 


790 


624100 


493039000 


28.1069386 


9.2443355 


791 


625681 


494913671 


28.1247222 


9.2482344 


792 


627264 


496793088 


28.1424946 


9.2521300 


793 


628849 


498677257 


28.1602557 


9.2560224 


794 


630436 


500566184 


28.1780056 


9.2599114 


795 


632025 


502459875 


28.1957444 


9.2637973 


796 


633616 


504358336 


28.2134720 


9.2676798 


797 


635209 


506261573 


28.2311884 


9.2715592 


798 


636804 


508169592 


28.2488938 


9.2754352 



264 TABLE OF SQUARES, CUBES, SQUARE AND CUBE BOOTS. 



Number. 


Square. 


Cube. 


.Square Root. 


Cube Boot. 


799 


638401 


510082399 


28.2665881 


9.2793081 


800- 


G40000 


512000000 


28.2842712 


9.2831777 


801 


CllGOl 


513922401 


28.3019434 


9.2870444 


802 


G43204 


515849G08 


28.3l9a045 


9.2909072 


803 


G44809 


517781G27 


28.3372546 


9.2947671 


804 


G4G416 


5197184G4 


28.3548938 


9.2986239 


805 


G4.S025 


521GG0125 


28.3725219 


9 8024 775 


80G 


G49G36 


523G0GG1G 


28.3901391 


9.3063278 


807 


G51249 


525557943 


28.4077454 


9.3101750 


808 


G528G4 


527514112 


28.4253408 


9.3140190 


809 


G54481 


5294 75129 


28.4429253 


9.3178599 


810 


G5G100 


531441000 


28.46049H9 


9.321G975 


811 


G57721 


533411731 


2S.4 780G17 


9.3255320 


812 


G59344 


53538 7328 


28.4956137 


9.3293(134 


813 


GG09G9 


5373G7797 


2S.5i:nr»J9 


9.3331916 


814 


GG2596 


539353144 


■-' ~ "'2 


9.33 701 G 7 


815 


GG4225 


541343375 


•J . ;s ^ 


9.34n.s386 


810 


GG585G 


543338496 


2.s.5(i:>7137 


9.344G575 


817 


GG7489 


545338513 


2S.5S32119 


9.3484 731 


818 


GG9124 


54 7343432 


28.6006993 


93522857 


819 


G707G1 


549353259 


2.*<.G1S17G0 


9.:^r}i]nn')2 


820 


G 72400 


5513(58000 




'■ »; 


821 


G 74041 


5533S7GG1 


1_ - , 


;!> 


822 


G75G84 


555412248 


2h.G7u^424 


9.3G75U51 


823 


G77329 


557441 767 


28.6879716 


9.3713022 


824 


G7.S97G 


5594 7G224 


28.7054002 


9.3750963 


825 


GS0G25 


561515625 


28.7228132 


9.3788873 


82G 


GS2276 


563559976 


28,7402157 


9.3826752 


827 


G83929 


565609283 


28.7576077 


9.3864600 


828 


G85584 


567663552 


28.7749891 


9.3902419 


829 


G87241 


569722789 


28.7923601 


9.3940206 


830 


G88900 


571787000 


28.8097206 


9.3977964 


831 


G905G1 


573856191 


28.8270706 


9.4015691 


832 


G92224 


575930368 


28.8444102 


9.4053387 


833 


G93889 


578009587 


28.8617394 


9.4091054 


834 


G9555G 


580093704 


28.8790582 


9.4128690 


SoO 


G97225 


582182875 


28.89G3G66 


9.4166297 


83(3 


G98896 


584277056 


289136646 


9.4203873 


837 


7U05G9 


586376253 


28.9309523 


9.4241420 


838 


702244 


588480472 


28.9482297 


9.4278936 


839 


703921 


590589719 


28.9654967 


9.431G423 


840 


705G00 


592704000 


28.9827535 


9.4353880 



TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. 265 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


841 


707281 


594823821 


29.0000000 


9.4391307 


842 


708964 


596947688 


29.0172363 


9.4428704 


843 


710649 


599077107 


29.0344623 


9.4466072 


844 


712336 


601211584 


29.0516781 


9.4503410 


845 


714025 


603351125 


29.0688837 


9.4540719 


846 


715716 


605495736 


29.0860791 


9.4577999 


847 


717409 


607645423 


29.1032644 


9.4615249 


848 


719104 


609800192 


29.1204396 


9.4652470 


849 


720801 


611960049 


29.1376046 


9.4689661 


850 


722500 


614125000 


29.1547595 


9.4726824 


851 


724201 


616295051 


29.1719043 


9.4763957 


852 


725904 


618470208 


29.1890390 


9.4801061 


853 


727609 


620650477 


29.2061637 


9.4838136 


854 


729316 


622835864 


29.2232784 


9.4875182 


855 


731025 


625026375 


29.2403830 


9.4912200 


856 


732736 


627222016 


29.2574777 


9.4949188 


857 


734449 


629422793 


29.2745623 


9.4986147 


858 


736164 


631628712 


29.2916370 


9.5023078 


859 


737881 


633839779 


29.3087018 


9.5059980 


860 


739600 


636056000 


29.3257566 


9.50968f54 


861 


741321 


638277381 


29.3428015 


9.5133699 


862 


743044 


640503928 


29.3598365 


9.5170515 


863 


744769 


642735647 


29.3768616 


9.5207303 


864 


746496 


644972544 


29.3938769 


9.5244063 


865 


748225 


647214625 


29.4108823 


9.5280794 


866 


749956 


649461896 


29.4278779 


9.5317497 


867 


751689 


651714363 


29.4448637 


9.5354172 


868 


753424 


653972032 


29.4618397 


9.5390818 


869 


755161 


656234909 


29.4788059 


9.5427437 


870 


756900 


658503000 


29.4957624 


9.5464027 


871 


758641 


660776311 


29.5127091 


9.5500589 


872 


760384 


663054848 


29.5296461 


9.5537123 


873 


762129 


665338617 


29.5465734 


'9.5573630 


874 


763876 


667627624 


29.5634910 


9.5610108 


875 


765625 


669921875 


29.5803989 


9.5646559 


876 


767376 


672221376 


29.5972972 


9.5682982 


877 


769129 


674526133 


29.6141858 


9.5719377 


878 


770884 


676836152 


29.6310648 


9.5755745 


879 


772641 


679151439 


29.6479342 


9.5792085 


880 


774400 


681472000 


29.6647939 


9.5828397 


881 


776161 


683797841 


29.6816442 


9.5864682 


882 


777924 


686128968 


29.6984848 


9.5900937 



23 



2GG TABLE OP SQUARES, CUBES, 8QU.UIE AND CUBE ROOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Boot. 


883 


779689 


688465387 


29.7153159 


9.5937169 


884 


78145G 


690807104 


29.7321375 


9.5973373 


885 


783225 


693154125 


29.7489496 


9.6009548 


88r, 


784996 


695506456 


29.7657521 


9.6045696 


8S7 


786769 


69r.S64103 


20.7825452 


9.6081817 


888 


788544 


700227072 


29.7993289 


9.6117911 


889 


790321 


702595369 


29.8 K, 1030 


9.6153977 


81)0 


792100 


704969000 


78 


9.6190017 


891 


793881 


707347971 


;i 


9.6226030 


892 


795664 


7079322H8 


'0 


9.6262016 


893 


797449 


712121957 


.,(] 


0.6297975 


894 


799236 


714516!)«4 


29>:'.--.-.'^ 


:'.«i;;:;::'*o7 


895 


801025 


716917375 


2M.r>l < ".'^M.; 


:».•.;;•.:»> 12 


8!M> 


802816 


719323186 


l 


l>. 6405690 


89 7 


804609 


721734273 


>3 


9.6441542 


898 


806404 


724150792 


1 


9.6477367 


899 


8U8201 


726572699 


^7 


9.6513166 


900 


810000 


729000000 


3o.<>ooouo0 


9.6548938 


901 


811801 


781432701 


80.0166621 


9.6584684 


•02 


813604 


7338 70H08 


30.0333148 


9.6620403 


903 


815409 


736314327 


80.0499584 


9.6656096 


904 


817216 


738763264 


30.0665928 


9.6691762 


905 


819025 


741217625 


80.O832179 


9.6727408 


90G 


820836 


743677416 


30.0998339 


9.6763017 


907 


822649 


746142643 


30.1164407 


9.6798604 


908 


824464 


748613312 


30.1330883 


9.6834166 


909 


826281 


751089429 


80.1496269 


9.6869701 


910 


828 100 


753571000 


30.1662063 


9.6905211 


911 


829921 


756058031 


80.1827765 


9.6940694 


912 


831744 


758550528 


30.1993877 


9.6976151 


913 


833569 


761048497 


30.2158899 


9.7011583 . 


914 


835896 


763551944 


30.2324829 


9.7046989 


915 


•837225 


766060875 


80.2489669 


9.7082369 


91(J 


839056 


76S575296 


30.2654919 


9.7117728 


917 


840889 


771095213 


30.2820079 


9.7158051 


918 


842724 


773620632 


30.2985148 


9.7188854 


919 


844561 


776151559 


30.3150128 


9.7223681 


920 


846400 


778688000 


30.3815018 


9.7258888 


921 


848241 


781229961 


30.3479818 


9.7294109 


922 


850084 


783777448 


30.3644529 


9.7329309 


923 


851929 


786330467 


30.3809151 


9.7364484 


924 


853776 


788889024 


80.3973683 

M 


9.7399634 



TABLE OF SQITAKES, CrBES, SQUABE AND CTJBE BOOTS. 26T 



j Number, 


Square. 


Cube. 


Square Ro©t, 


Oube Koot. 


r 

j 925 


855625 


791453125 


30,4138127 


^.7434758 


1 926 


857476 


794022776 


30.4302481 


9,7469857 


1 927 


859329 


7^6597983 


30.4466747 


9.7504930 


1 928 


861184 


799178752 


30.4^30924 


9.7539979 


; 929 


863041 


•801765089 


30.4795013 


9.7575002 ; 


930 


864900 


80435 7000 


30.4959014 


9.7610001 


931 


866761 


80<595449i 


30.5122926 


9.7644974 


932 


868624 


80955 7568 


30,5286 750 


9.7679922 


933 


870189 


812166237 


30.5450487 


9.7714845 


934 


672356 


814780504 


30.5614136 


0.7749743 


935 


874225 


817400375 


30.5777697 


9.7784616 


93G 


876096 


820025856 


30.5941171 


9.7829466 


937 


877969 


822656953 


30.6104557 


0.7854288 


938 


879844 


8252936 72 


30.626785 7 


9.7889087 


939 


881721 


827936019 


30.6431069 


9.7923861 


940 


883600 


830584000 


30.6594194 


9.7958611 


941 


885481 


833237621 


30,6757233 


0.7993336 


942 


887364 


835896888 


30.6920185 


9.8028036 


943 


889249 


838561807 


30.7083051 


9.8062711 


944 


891136 


841232384 


30.7245830 


9.8097362 


945 


893025 


843908625 


30.7408523 


9,8131989 


946 


8t)49l6 


846590536 


30.7571130 


9.8166591 


• 947 


896808 


8492 78123 


30.7733651 


9.8201169 


948 


898704 


851971392 


30.7896086 


9.8235723 


949 


900601 


8546 70349 


30.8058436 


9.8270252 


950 


90250<^ 


857375000 


30.8220700 


9.8304757 


951 


904401 


860085351 


30.838287^ 


0.8339238 


952 


906304 


862801408 


30.8544972 


9.8373695 


953 


908209 


865523177 


30.8706981 


9.8408127 


954 


910116 


868250664 


30.S868904 


9.8442536 


955 


912025 


870983875 


30.9030743 


9.8476920 


956 


913936 


873722816 


30.9192477 


9.8511280 


1 957 


915849 


876467493 


30.9354166 


9.8545617 


958 


917764 


879217912 


30.9515751 


0.8579929 


959 


919681 


•881974079 


30.9677251 


9.8614218 


960 


921600 


884736000 


30.9838668 


0.8648483 


961 


023521 


887503681 


31.0000000 


9.8682724 


962 


925444 


890277128 


31.0161248 


0.8716941 


963 


927369 


893056347 


31,0322413 


9.8751135 


964 


929296 


895841344 


31.0483494 


9.8785305 


965 


931225 


898632125 


31.0644491 


9.8819451 


966 


933156 


901428696 


31.0805405 


9.8853574 



268 TABLE OP SQUARES, CUBES, SQUARE AND CUFB ROOTS. 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


967 


935089 


904231063 


31.0966236 


9.«8«767.S 


D68 


937024 


907039232 


31.1126984 


( , -If 


969 


938961 


909853209 


31.1287648 


1 


070 


940900 


912673000 


31.1448230 


;■ , 


971 


942841 


915498611 


31.1608729 


; - , ;, 


972 


944 784 


9183300J8 


81.1769145 


[i.'Mtiil si 7 


973 


946729 


921167317 


31.19294 79 


9.9091776 


974 


948676 


924010424 


31.2089731 


9.9125712 


975 


950625 


926S59375- 


31.2249900 


9.9159624 


976 


952576 


929714176 


31.2409987 


9.0193513 


977 


951529 


932571833 


31.256I»992 


9.0227370 


978 


956484 


935441352 


31.2729915 


9.9261222 


979 


958141 


938313739 


31.2.S89757 


I».0205o|2 


980 


960400 


941192000 


31.304Ii517 


9.IK'J2.'^>'.I!> 


981 


962361 


944076141 


31.3209195 


9.0362<;i:{ 


982 


961324 


946066168 


31.3368792 


9.9306;;<;;; 


983 


966289 


949^62087 


31.3528308 


9.0430O!»2 


984 


968256 


952763904 


31.3687743 


9.0163707 


985 


970225 


955671625 


31.3.H4 7097 


9.9497479 


986 


972196 


958585256 


31.4006369 


9.0531138 


987 


971169 


961504803 


31.4165561 


0.0564 7 75 


988 


976144 


964430272 


31.43246 73 


9.059S3SJ) 


989 


978121 


967361669 


31.44H3704 


9.0631 081 


990 


980100 


970299000 


31.4642654 


9.0665540 


991 


982081 


973242271 


31.4801525 


9.9699095 


992 


984064 


976191488 


31.4960315 


9.9732610 


993 


986049 


979146657 


31.5119025 


9.9766120 


994 


9H8036 


982107784 


31.5277655 


9.9799599 


995 


990025 


985074875 


31.5436206 


9.9883055 


996 


992016 


98804 7936 


31.5594677 


9.98664S8 


997 


994009 


991026973 


31.5753068 


9.9899900 


998 


996004 


994011092 


31.5911380 


9.9988289 


999 


998001 


997002999 


31.6069613 


9.9966656 , 


1000 


1000000 


1000000000 


31.6227766 


10.0000000 


1001 


1002001 


1003003001 


31.6385840 


10.0088222 


1002 


1004004 


1006012008 


31.6543836 


10.0066622 


1003 


1006009 


1009027027 


31.6701752 


10.0099899 


1004 


1008016 


1012048064 


31.6859590 


10.0133155 


1005 


1010025 


1015075125 


31.7017349 


10.0166380 


1006 


1012036 


1018108216 


31.7175030 


10.0109601 


1007 


1014049 


1021147343 


31.7332633 


10.0232791 


1008 


1016064 


1024192512 


31.7490157 


10.0265958 



TABLE OF SQUARES, CT7BES, SQUARE AXB CUBE ROOTS* 269 



Number, 


Square, 


Cube. 


Square Hoot. 


Cube Root. 


1009 


1018081 


1027243729 


31.7647603 


10.0299104 


1010 


1020100 


1030301000 


31.7804972 


10.0332228 


1011 


1022121 


1033364331 


31.7962262 


10.0365330 


1012 


1024144 


1036433728 


31.8119474 


10.0398410 


1013 


1026169 


1039509197. 


31.8276(509 


10.0431469 


1014 


1028196 


1042590744 


31.8433666 


10.0464506 


1015 


1030225 


1045678-375 


31.8590646 


10.0497521 


1016 


1032256 


1048772096 


31.8747549 


10.0530514 


1017 


1034289 


1051871913 


31.8904374 


10.0563485 


1018 


1036324 


1054977832 


31.9061123 


10.0596435 


1019 


1038361 


1058089859 


31.9217794 


10.0629364 


1020 


1040400 


1061208000 


31.9374388 


10.0662271 


1021 


1042441 


1064.332261 


31.9530906 


10.0695156 


1022 


1044484 


1067462648 


31.9687347 ' 


10.0728020 


1023 


1046529 


1070599167 


31.9843712 


1-0.07^0863 


1024 


1048576 


1073741824 


32.0000000 


10.0793684 


. 1025 


1050625 


1076890625 


32.0156212 


10.0826484 


1026 


1052676 


1080045576 


32.0312348 


10.0859262 


1027 


1054729 


1083206683 


32.0468407 


10.0892019 


1028 


1056784 


1086373952 


32.0624391 


10.0924755 


1029 


1058841 


1089547389 


32.0780298 


10.0957469 


1030 


1060900 


1092727000 


32.0936131 


10.0990163 


1031 


1062961 


1095912791 


32.1091887 


10.1022835 


1032 


1065024 


1099104768 


32.1247568 


10.1055487 


1033 


1067089 • 


1102302937 


32.1403173 


10.1088117 


1034 


1069156 


1105507304 


32.1558704 


10.1120726 


1035 


1071225 


1108717875 


32.1714159 


10.1153314 


1036 


1073296 


1111934656 


32.1869539 


10.1185882 


1037 


1075369 


1115157653 


32.2024844 


10.1218428 


ioa8 


1077444 


1118386872 


32.2180074 


10.1250953 


1039 


1079521 


1121622319 


32.2335229 


10.1283457 


1040 


1081600 


1124864000 


32.2490310 


10.1315941 


1041 


1083681 


1128111921 


32.2645316 


10.1348403 


1042 


1085764 


1131366088 


32.2800248 


10.1380845 


1043 


1087849 


1134626507 


32.2955105 


10.1413266 


1044 


1089936 


1137893184 


32.3109888 


10.1445667 


1045 


1092025 


1141166125 


32.3264598 


10.1478047 


1046. 


1094116 


1144445336 


32.3419233 


10.1510406 


1047 


1096209 


1147730823 


32.3573794 


10.1542744 


1048 


1098304 


1151022592 


32.3728281 


10.1575062 


1049 


1100401 


1154320649 


32.3832695 


10.1607359 


105G 


1102500 


1157625000 


32.4037035 


10.1639636 



25* 



270 TABLE OF SQUARES, CUBES, SQUIKE AJTD CUBE ROOTS. 

NOTF. — Since all numbers aro the square rofifs of tluir Founrrs, and the 
cube roots of (heir cube.s, ft follows that the ininiUrsi tabwlate*! art the sijuare 
jools of thtir respective' si(|uures^ and tlio cube roots of their respective cubes. 
3Ioreover, one-half, one-third, one-fourth, &c., of the sijuari' root of any num- 
ber is the fiquare root of (/ne-fourti), mic-uinth, one-nixteentli, &c., cif that 
number; thius one-tenth of the s(|uj»rc nx* of KiO is the scjuarc nn^t of 150» 
-r- 100 = VHt ftnd one-lourth part of the i^tiuarc root of iJo.T'i, is the Hjuare 
root of 1J.'>.72-^ 1<*» = v^.s>;5, &c. lUit theprocfss of extractin>r tin* sjpiure 
roots of nunib<'rs by arithmitic is so simple, and the lalior »o trifliuf?, that 
but little will he Kain« d, piin.r^illy, by rvsorlvng to the tubleu, copcxially VtheB 
the square root nt' a mixed number is required. 

Br/ the help of the Table^ to find the Crdte Root of a number that is 
not the exact Cube of am/ nirmher tahidatcd, and that is nfti f/reat- 
er than the Cube of one more than the h'erjhrst number tidfulated. 
n =z rrjvcn number whose cube root is reqairecL 
<=^ tabular cul>c nearest the given number. 
b z=z tabular cube root of /. 
r = cube root of given niunbcr, or cube root rcf^uircd. 

NoTK. — This formula cxprr^«r«» thf- rube mot mrrert tn withfn nhmit 
l-l(H)of n unit, undrr its m<»**t uiiTi . i p»<t 

•trict accurary, cnch wlun th** n .-r 

br JarK*'. Wlirn t wo tal>ular ml" . ' ry 

nearly 80, use the ffreator, by which a cloacr approxiuiaU will U; ubluiiit^U, aad 
r will be plus by the amount of the error. 

Bt/ help of the Tdblcy to find the Sf/tiare Root or Cube Root of a 

mixed number^ whose integer^ and thr integer next higher, are tain 

ulated, 

n = given mixed number whc^^e i.p..i is roquirod. 

R =z root of the integer next high(T than the integer of tho 
given num?)er. 
i = integer of the given number. 

.s = root of the integer of the given nuitiixT. 

r HZ root of the given mixed number, or root required, 
r = (72 — s)(ii — i) -\- -s practically correct for ordinary purposes. 

Example. — Required the sf|uarc root and cube root of 18.54. 



V/1 9=: 4.358898a 

V^18 = 4.2426407 

.1162582 

.54 



4650328 
.5812910 



.062779428 
V^18 = 4.2426407 
V^18.54 = 4.305420128. Ans. 



Vl 9 =2.668401(5 
Vl8 = 2.6207414 





.0476602 
.54 




VI 8 = 
V18.54 1= 


1906408 
2383010 
.025736508 
12.6207414 
2.646477^08. 


Ans. 



SECTION V. 

MECHANICAL POWERS, CIRCULAR MOTION, &c. 



i 



The Mechanical Powers are the know^n elements of machiiiery. 
They are three in number, with some diversity of application. 
Strictly speaking, they are not powers, or sources of power ; they 
simply convey applied force, and diffuse or coiycentrate it. In treat- 
ing of them, the term loeigJit, or resistance, is understood to be the 
force to be overcome, and the term power, the force applied to over- 
come or balance it. It is also to be understood that the deductions or 
conclusions arrived at are theordicaUy true ; that is, that they are true 
upon the supposition that the whole power employed is expended to 
the end under consideration — that no friction or weight of machinery 
aids, or is to be overcome. 

THE LEVER. 

Lemma. — The power multiplied by its distance from the fulcrum 
equals the weight multiplied by its distance from the same point ; and 
as the distance between the power and fulcrum is to the distance 
between the weight and fulcrum, so is the effect to tlie power. 

Consequently, if we divide the weight by 
the power, we obtain a quotient equal the length 
of the longest arm of the lever, the length of 
the shortest arm being 1. And if we multiply 
m/" the weight by its leverage, and divide the prod- 

^ -uct by the power, we obtain a leverage for the 

power that will enable it to equipoise the 
weight. And if we multiply the power and 
its leverage together, and divide the product by 
the weight, w^e obtain for the weight the same 
<i result. So, too, if we divide the lever by the 
quotient obtained by dividing the weight by the power, to which quo- 
tient we have added 1, we obtain the relative position of the fulcrum, 
or the distance it must occupy from the opposing force. And, again, 
if we multiply the opposing force by its leverage, and divide the prod- 
uct by the leverage pertaining to the power, we obtain the requisite 
power to counterbalance the resistance. 

Example. — A weight of 1200 lbs., suspended 15 inches from the 
fulcrum, is to be raised by a power of 80 lbs. ; at what distance from 
the fulcrum on the long arm of the lever must the power be applied, to 
accomplish that end ] 

80 : 1300 :: 1.25 : I8i feet. Ans, 



T 



272 



KECIIAKICAL POWERS, 



Example. — The lever is 20 feet long, the opposing forc^J 
3200 lbs., and the available force 80 lbs. : at what distance from th«1 
ibrmer force must the fulcrum be placed, that the two forces may J 
equipoise each other ? 20 X 80 -f- (1200 -f- 80) = I J feet ; or 

1200 ~- 80 = 15, and 20 -^ 15+ 1 » l\ feet. Ans. 

Example. — The longer arm of the lever is 18} feet, the sliorti 
arm l.i feet, and the \yeight to he raised is 1200 lbs. ; what powerl 
must Im; a[)' lied lo raise it? 

18.75 : 1.25 :: 1200 = 80 lbs. Ans. 

Example. — A nrKin, with a lever 5 feet iu length, raided a weight 
of 2500 lbs. siispendcil acroe>3 the lever 9 ir>cbes from the further end, 
which rested on a .support ; what force did the nyin e.xert ? 
5 : .75 :: 2500 = 375 lbs. Ans. 

Example. — A beam, 20 feet in length, supported at both ends and 
not elsewhere, Ix^ara a weight of 0000 ll/s. placed 6 feet from one 

end ; what is the pre.'»«ure on each sup|x>rl ? 

20 III:: noon ; IJim Ihs. on the supi»ort nearest the weight. > . 

20 : G :: OOOO : loOO lb». on the support furthest from the w'l. \ 



Ans, 




WHEEL A.ND AXLE. 

The WHEEL and axle is a revolvimr lever. It 
partakes, in all respects, of the s:une principles 
as the preceding. The radius of the wheel is 
the longer arm of the lever, and the nulius <)f 
the axle, the shorter. The fulcrum is the point 
of impact between them — ■-* ♦'• ■ ^" ■rnference 
of the axle. 

Example. — The radius of ihc wUcv] is 2i 

feet, the radius of the axle is 9 inches, and the weirrht to be nnsrd ia 

500 lbs. ; the weij^Hit is attached to a rope wound round the axle ; 

what power must l>o applied to the periphery of the wheel to raise it* 

2.5 : .75 :: 500 : 150 lbs. Ans. 

Example. — The diameter of the wheel is 5 feet, the dinmner ot 
the axle or barrel, l-i feet, and the power is 150 lbs. ; what weight 
may be raised ? 

1.5 : 5 :: 150 : 500 lbs. Ann, 

Example. — The power is 150 lbs., the resistance is 500 lljs., and 
the barrel has 9 inches radius; recjuired the dinm( t'-r tf lli.- whn'l 
that will enable the power to equipoise the weight. 
150 : 500 :: 9 : 30 in. radius, and 30 X 2 =00 -J- 1. _ .> ., . .. ,.,,,. 

Example. —The length of the winch (crank) of a crane is 15 



MECEANICAl POWERS, 



273 



inches, the radius of the barrel around which the lifting chain coils ifl 
3 inches, the pinion has 8 teeth, and the wheel 68; requir^ the 
weight that a force of 30 lbs. applied to the winch will raise. 

68 -~ 8 = 8J (8i to 1) velocity of pinion to wheel, and 
15 X 8.5 -7- 3 = 42.5 lbs. exertive force, or force to 1 of applied 
power — gained at the expense of space, and 
42.5 lbs. X 30 lbs, (applied power) = 1275 lbs. effective power. Ans. 

Example. — The exertive force, or effect to power, of a crane, is 
to be as 42i to I , the radius of the wheel to that of the pinion as 8i 
to 1, and the throw of the winch — its length — the radius of the cir- 
cle which it describes — is to be li feet ; what must be the diameter 
of the barrel ? 

8.5 X 1.25 = 10.625 — 42.5 = .25 X 2 = .5 ft. or 6 in. Ans, 

Note. — By additional wheels and pinions, as in the system of pulleys or block an<l 
tackle, which see, the exertive force of a crane may be increased to almost any conceiva- 
ble extent ; but always, as with the Uock and tebckle, aftd as showa in Um above i 
pie, at a relative expense to spax:e. 




A single pulley, fixed and turning on its own axis, affords no me- 
chanical advantage. It serves but to change the direction of the 
power. 

In the common system of pulleys, or block and tackle, the advan- 
tage is as the number of ropes engaged in supporting the lower or 
rising block, to 1 of applied force. 

Rule. — 1. Divide the given weight by the number of cords lead- 
ing to, from, or attached to, the lower block, and the quotient is the 
requisite power to produce an equilibrium. 

Rule, — 2. Multiply the given power by the number of cords lead- 
ing to, from, or attached to the lower block, and the product is the 
weight that may be raised. 

Rule. — 3. Divide the weight to be raised by the power to be 
applied, and the quotient is the requisite number of cords that must 
connect with the lower block. 

Exaaiple. — The lower, running, or rising block has 5 sheaves or 



vet ■! 
bodff 



274 MECHANICAL POWERS. 

pulleys ; the fixed or stationan' has 4 ; and the wei|rht to be raised 
is 2250 lbs. What force mnst'he applied to raise it? Necessarily 
the end of the rope is attached to the lower block, therefore 9 ropes 
are attached to or connected with it ; hence — 

2250 -^ 9 =* 250 lbs. Ans. 

Note — In il>e Spanhh burton, having two roomble pulleys wkI iwo aepante ropca, 
the ctTect id I" iho |»wor aj 5 to I. In a ayslem of 4 movable oulleys and 4 separata 
ropes, it isaa IT. to I. Ami in a HYMU?m havinir 4 movable a[ul4 fixed puUeya, and 4 mi>> 
arate ropes, it is as SI to 1 

INCLINKD rLA:<E. 

Lemma.— T\\c product of the length of the plane and power 
equal to the product of the height of the plane and weipht. 

TIic velocity, therefore, or force, or rnoinentiim, with which a 1 
descends an inclined plane, impelled by its own gravity, is tti that 
with which ihe. feaiuc body would descend pcrpcndicularh tbrnntrh 
space, as the bright of the plane to its Iciifrih, or as thr 's 

angle of inclinatiun to radius. And the space the body desu ,00 

tlie plane, in any given time, compared with that which it would de- 
scribe falling frrdy, in the same time, is as its velocity upon the plan*- 
to that of perpcnd'iculiir descent. And, the spaces being the 8;imr 
the times will bo invrrsely in that proportion. 

The dcduciiuns, therefore, are — 

1. That the ]»roduct of the weiirht and height «>• j-.-.- , *iividcd I 
the length of plane, gives the requisite power to sustain or balanr. 
the weight. 

2. That the prodiict of the power and length of plane, divided b 
the height of plane, (which reverses the former process,) gives th 
weight or resistance that the power will overcome. 

3. That — (the times being ec]ual) — the velocity attiined, <• 
force acquired, or space described, by a body falling freely from res' 
multijdicd by the height of plane, alTords a product which, divided bv 
the length of i)lane, gives the velmnty attained, or force acrpiired, or 
space described, by a body moving down the plane, impelled by its 
own gravity. 

I. That the product of the weight and base of plane, divided by 
the Icnglii of plane, gives the pressure on the plane. 

To find the base of the phne. 
R^LE. — From the square of the length of the plane, subtract the 
square of the height, and the square root of the ditrorence is the ba^e. 

To find the height of the plane. 
Rule. —From the square of the length of the plane subtract ll. 
square of the base, and the square root of the ditference is the height 



MECHANICAL POWERS. 



275 




To find the length of the plane. 
Rule. — Add the square of the base and the square of the height 
tog-ether, and find the square root of the sum, which will be the 
length sought. 

WEDGE. 

The WEDGE is a double inclined plane. Its principles are the same, 
and they are wholly covered by the preceding. 

The pov/er multiplied by the length of a side, equals 
the resistance multiplied by half the breadth of the head. 

When, therefore, both sides of the substance to be 
cleft are movable, the product of the resistance and half 
the breadth of the head, divided by the length of the side 
of the wedge, gives the requisite force to be applied. 
And when only one side of the substance is movable, the 
product of the resistance and breadth of the head, divided 
by the length of a side of the wedge, gives the power 
required. 

SCREW. 

If we take the figure of an inclined plane — a 
right-angled triangle say, cut from paper — and 
unite the extremities of the base, we have the 
figure of the screw, in principle ; and the prin- 
ciple of the screw is that of an inclined plane 
curved to a cylinder ; and the screw is not a me- 
chanical power, any more than the wedge, or 
wheel and axle. It is the plane that is an element 
I of machinery, and not the curve or the cylinder 
I around which the plane is placed. And it appears 
that the screw, the incliiied plane, and the right- 
"1^ \ angled triangle, are mathematically the same. 
Thus, if we would find the length of the thread of a screw by the 
circumference and pitch, we are to find it as we would find the 
length of the inclined plane by the base and height, or the hypote- 
nuse of a triangle by the base and perpendicular, and so, in like 
manner, for the other lines of the figure. 

The pitch of the screw or rise of the thread in a revolution corre- 
sponds to the height of the plane or perpendicular of the triangle. 
The circumference of the screw corresponds to the base of the plane 
or base of the triangle. And the length of the thread making one 
revolution around the cylinder — the loorhing circumference of the 
screw— corresponds to the length of the plane or In-potcnuse of the 
triangle. The mechanical advantage of a screw is as the length of 
the plane to tl)e size of its angle of inclination. 




276 MEcniyiCAL powkw. 

The ordinary scmo, therefore, — the piece of mechanism, — is thi» 
plane repeated along a cylinder a greater or less numl>er of times, 
whereby the spiral thread alluded t4), now the sectional thread of 
the instrument, lx?coine» the confhiuoiis thread, helix or spiral, thai 
extends, by construction, at a unifunii angle to the cylinder's axis, 
throughout its length. And it is to }>e lx)me in mind, always, that 
the pitch of a screw is the distance, parallel to the axis, hotweon 
any two consecutive Avindings of the Mme apparent continuous 
thread, measured on its face, from centre to centre. 

If a scn'w liave more than one apparent continuous thread, there- 
fore, — and screws are often constructed with two, and sometimes 
more — the pitch of such screw is still the samo as it would Ixj wcro 
but one such thn^ad employed, or as it would 1x5 were all Init one 
removed. Twelve rafters to the side of the roof of a Ixiilding can 
sustain more pressure than four, and in the ratio of three to one, but 
the pitch of such rcxjf is neither greater nor less in consoquonoc of 
the number of rafters eroployeii. 

The screw multiplies the extent of the action of the inclinH plane, 
it will be perceived, as many times as the plane is r 1 mg 

its cylinder, whereby great advantige oi" meclianical _ u is 

obtiiintxl ; but the in/rhankal advantoi^r of the screw, it is apjuirent, 
is still with the plane, and not in the numlK3r of times the plane is 
used. The towkr is no more with the latter than it is with tho 
curve of the plane (which is another advantiigo of mechaniuvl appLi 
cation^ instead •>f U^ing in tlie plane itself. 

And there is still another advantage }Hirtiiining to the screw, or tr 
this hkhIo of employing the incliiicd plane, viz., tliat the length of 
the plane or working circumference of the screw may lie iin r. is.d, 
in efiect, to alm<;8t or quite any desin'd extent, by the rnij 
of a simple Nir or rod to turn tho s<row ; whereby, the ;> 
j>itrh remaining the Siime, the mechanieal iulvantige is enhanced as 
much as the working circumference is increased ; that is, it is made 
greater i\s much as the circumference of a circle, the bar being the 
radius therei>f. 

The foxccr^ftyrccj or mechanical a/irontat^e of a screw, as we have 
said, is that of the inclined plane employed, and is as its length to 
the size of its angle of inclination, or it is as the working circumfer* 
ence of the screw to tlie pitch. 
K we let P represent power, 

L ** length of lever, 

/ ** length of inclined plane, 

W " weight or pressure, 

p '* pitch of screw or angle of inclination, 

r ** radius of screw, 

C '* circumference de8cril)ed by power, 

X '* elfcct of power at circumference of screw 



MECHANICAL P0WEE3. 277 



Then we have — 
lip ::^Y :V 
Z : W ::p : P 
P : W ::^ : / 



W :I ::V :p 
r :L :: F : X 
F : X :: r :L 



L : r : : a? : P. 
V :W ::p :G 
C :p ::W iF 



Example. — The circumference of a screw is 12 inches, its pitch 
1^ inches, and the power is 30 lbs. ; what weight may be raised 1 

/v/(12" -|~ li") == 12.065, working circumference of screw, and 
1.25 : 12.065 : : 30 : 289,56 lbs. Ans, 

And if a bar 14 feet in length (rectilinear distance from the 
point on the bar at which the power is applied to the circumference 
of the screw) be employed to turn the screw, the power remaining 
the same, what weight may be raised? 

14 X 12 X 2 X 3.1416 = 1056 inches, circumference of circle, 
bar as radius ; and 

1056 -|- 12.065 = 1068.065 inches, circumference described by 
power ; and 

1.25 : 1068.065 : : 30 : 25633 lbs. Ans. 

Or, 12.0G5 : 289.56 : : 1068.065 : 25633 lbs. Ans. 

The foregoin<^exhibits the method of finding the strict theoretical 
force or mechanical advantage of the screw. But for most practical 
purposes, more especially if wo take into account the fact that the 
actual force in consequence of the friction is only about two-thirds 
that of the theoretical, the rectilinear distance from the point on the 
bar at which the power is applied, to the centre of the screw, may 
be taken as the radius of the circumference described by the power, 
or as the elfective or working circumference of the screw, instead of 
the true working circumference as found above ; and this is more espe- 
cially true if the pitch of the screw be but slight or inconsiderable. 
Thus, in the aforementioned screw of 12 inches circumference and 
1\ inches pitch, the difference between the actual circumference and 
the working is only .065 of an inch, which, when the effect is of some 
magnitude, is of no particular account. The example next below is 
illustrative, and is given in proof of this position. 

Ex^\MPLE. — The pitch of a screw is li inches, the power 30 lbs., 
and the rectilinear distance from the centre of the screw to the point 
on the bar at which the power is applied is 169.91 inches ; required 
the weight that may be raised, supposing this rectilinear distance to 
be the radius of the circle described by the power. 
24 . 



278 MECHANICAL POWERS. 1 

1G9.91 X 2 X 3.1416 ="1067.57 inches, circumference by asTOmed 
radius ; and 

1.25 : 1067.57 : : 30 : 25621 lbs. Ans, And showing an errot 
of only 12 lbs. in 25633 lbs., consequent upon having employed the 
assumed radius instead of the real, and that, too, under a pitch so 
unfavorable as the one supposed. 

AVhon a hollow screw revolves upon one of less diameter and. 
pitch, the cfloct is the same as that of a single screw whose pitch i 
equal to the difference of the pitches of the two screws. 

Thus, if a hollow screw of ^ of an inch pitch revolves upon on 
of ^ of an inch pitch, the power to the weight is as | -^ ^ = y^j ;' 
that is, the ]K>wcr l>oing 1, the weight will 1x3 24. 

A screw of this description and with those pitches, therefore, if ■ 
turned with a bar 6 inches in length, (distance fnmi the iM»wer tol 
the centre of the screw,) will, in onkr to pHnluco an oquilibrium, 
require a fK>wer to the weight as 1 to 24 X 2 X 3.14ir) X 6 = 905. 

in a complex machine, composed of the screw and wheel and axle, 
the relalions of the weight and power are aa under : — 



ooel 



Let K reprc,-*cnt r;»«liu(4 "f wlvol. 
r " r.i'liuH ('( axle. 



p '• piich of screw. 

C " circumforcucc dcscril«il \rr power. 

X " etftcl vf puwor on ihe wheel. 

Tb«i- 



PXCXK=WXpXr 
P : W : : p X r : C X i: 
PX r:CXK::t: W 



4 



PXC = T XP 
W X r = K X ^ 
PXCX'XR = 'XWx»'XP 
And, if intermediate movers arc inserted, (wheels and pinion*,' 
drums and pulleys,) the same principles slill apply. 

Example. — The lenglh of the crank (le^er) which turns an end- 
less screw, is 24 inches, and the pilch of ihe screw is } of an inch; 
it turns a wheel of 30 inches radius, which turns a pinion of 7 inches 
radius, which turns a wheel of 22 inches radius, which turns an axle, 
around which the lifiing chain winds?, of three inches radius; what 
weight will a power of 50 lbs. applied to the crank raise? 
p r r C R R P W 

.75 X 7 X 3 : 150.8 X 30 X 22 :: 50 : 3151)62 Ibe. Arts. 

NoTB. — It is clear, wc may sub^tiitite the diameters of the wheels and plnioM fbf 
their radii, if we prefer ; or we may work by ihc number of teelh in each, in which latttf 
case, the circumference of llie axle, in the foregoin;?, would comc in to be employed. 

If the screw, acting upon the periphery of the wheel, have more 
than one thread, the real, or ohvious pitch, spoken of above, must be 
taken, increased an equal number of limes. 



LATERAL OR TRANSVERSE STRENGTH OF BODIES. 



279 



LATERAL OR TRANSVERSE STRENGTH OF BODIES. 

The transverse strength of a body is its power to resist force or 
weight acting upon it in a direction perpendicular to its length. 

Against each particular denomination of material in the following 
TABLE is placed the weight (mean of various experiments) required to 
break a solid, uniform bar, One Foot in Length and One Inch Square, 
of that material, the bar being fixed at one end and the weight sus- 
pended from the other, the action of the weight direct with the bar's 
sides. 

The Woods of American growth, and seasoned. 





Breaking 


Greatest 




Breaking 


Greatest 


Materials. 


Weiglit in 


Deflection 


Materials. 


Weight in 


Deflection 




lbs. 


in Inches. 




lbs. 


in Inches. 


Hickory, 


270 


8. 


Pitch Pine, 


225 




White Ash, 


234 


2.5 


Yellow Pine, 


150 


1.70 


White Oak, 


220 


9. 


White Pine, 


138 


1.40 


Chestnut, 


170 


1.7 


Cast Iron, 


684 


0.63 


Elm, 


142 




Wrought Iron, 


1012 





About 600 lbs. suspended from the end of a square bar of wrought 
iron, of dimensions and fixed as supposed, causes the bar to deflect 
about one inch, at which it takes a permanent set, or bend. 

As the weight written against any particular denomination of ma- 
terial, in the foregoing table, is the weight required to break that 
material, under the length, lateral figure, and condition supposed, it 
follows that the same weight may be taken as the constant or co- 
efficient in determining the weight required to break the same denom- 
ination of material, under different lengths, lateral figures, relative 
conditions, &c. 

C = tabular constant, or initial weight, above. 
I = length of bar or beam in feet. 
b = breadth of rectangular bar in inches. 
V = vertical dimensions, or depth of rectangular bar in inches. 
^y = breaking weight, or ultimate transverse strength of bar under 
investigation. 

1 . When the bar is fixed at one end, and the weight suspended from 
the other ; the weight of the bar not being taken into account, 

Z:?;-X^ :: C : W. 

2. When the beam is supported (not fixed) at both ends, and the 

weight in the middle. 

Z : ^2 .. 4Q . ^ 

3. Fixed at both ends, and the load in the middle, 

/ : Z>i/^ :: 60 : W. 



280 LATERAL OE TEANSYSBSB STRENGTH Of BODDB. 

4. Fij^d at one end^ and the load distributed uniforrr^h '^•"t 
whole length, 

l'.v'h:'.2C: W. 



4 



5. Supported at both ends, and load distributed umformly over whole 
length . , 

liv'b:: 8C : W. 

G. Supported at both ends, and load at the distance mfrom one end, 
m X (/ — '") : I'W:: C : W. 
Or, 2(1 — 711) X -(/ — w) -i- / = /» = effective length, and 
/• : hv' :: 4C : W. 

NoTK. — bv^ In a square liciiin = a 8i«le of the square )>cain cubed. 

KxAMi'LE. — A l)c;im of white Oak, 3 fcN't in length, 4 inches deep, 
an«i 2 inch(« in hn^ailth, i« HuhI at one end ; \Yhat weight id ri'tjuired 
to break the b»*am, the weight l>eing suspended from the uther «.'nd? 

a : 4« X 2 :: 220 :: 2340i Ibe. Ans. 

Example. — ^The same l)eiun, same manner of 8up|X)rt, itc, a« the 
foregoing, hut the greater cruflp-ecction of the iH-ain placfil horizon- ' 
tally ; what weight is required to break it ' 

3 : 2^X4 :: 220 : ll73i iba. Ans. j 

KxAMri.E. — A l)eam of cast iron, 8 feet in length and G inches J 
Hrpiare, is supportinl at Ixith ends ; what is its ultimate transverse 
strength, it In'ing loaded in the middle? 

8 : G' X <>, I. <?., 8 : 6^ ;: 684 X 4 : 73872 Ibe. Ans. 

A l>oam fi.TtM] at both <'ndH, othor things Ix^ing enual, will bear OOd^ 
half nioro than wh«»n merely nupport^'d at l>oth enus. 

The h»ngth of a lx»am Hupportod at each end, or by two supports, is 
the distance from one Hnp|M)rt to the otiier. 

l^ound beams, 8up|>orted in the mi«ldK» and loaded at each end, 
have the wimo siistaniing ]>owor as when supported at each end and 
loadeil in the middle ; and the same is true for rectangular lx»ums, tho 
action of the load, in lujth cases, iKMng, in tho same manner, direct 
with the lH»am'8 central plane. 

An ecpiilateral triangular l>eam, supported in the middlo, and 
loadtnl at eaeh end, has the same sustaining power Jis when supported 
at each end and loaded in the middle, if the beam bo inverted. 

When a beam is partly loaded at any given locality of its length, to 
find what ir right, acting upon it at any other given locality, must be 

added to hrak it. 

Lot E = breaking weight at the locality of the given partial load. 
F = brciiklug weight at the locality of the required partial load. 



LATEEAL OR TRANSVERSE STRENGTH OF BODIES. 281 

Let a = given partial load. 

X = required partial load ; then 

E : F :: E— -a : oo. 

With regard to required depths, breadths, Sfc, 

Let S = effective coefficient in all cases ; that is, = C, or any mul- 
tiple of C, demanded by the conditions, as set forth in Prob. 1, 2, 3, 
4, 5, or 6, foregoing ; then 



/ : v'h :: S : W. 

W : S :: Z^u^ : L 



^^ :W ::l:b. 
8b:W iiliv". 



ji/ ( I = V, and --- = 6; _ being the ratio fixed upon for the 

\ SZ>^ / 1)" b^ 

depth to the breadth. 

Example. — What must be the depth of a pitch-pine beam, resting 
on tvro supporters 20 feet apart, that its ultimate transverse strength 
may be 24000 pounds, the beam being 4 inches in breadth, and the 
load resting uniformly along its whole length ? 

By referring to the table of initial weights, we find the prime co- 
efficient for pitch pine to be 225 lbs., and by problem 5 we find that 
8 times that quantity, or 1800 lbs., is the effective coefficient for the 
case in hand. Hence 

1800 X 4 : 24000 :: 20 : V66| = 8.165 inches. Ans. 

Comparative transverse strength of figures, or of beams, <^c., of differ- 
ent figures and positions ; both members of the couplet supposed to be 
of the same material, same length, in the same manner loaded, and in 
the same way supported, 

D, a square beam, the weight acting direct with the sides. 

O) a square beam, the weight acting direct with the diagonal. 

A J an equilateral triangular beam, the weight acting direct with 
the perpendicular, and tending to convex a side. 

V, an equilateral triangular beam, the weight acting direct with 
the perpendicular, and tending to convex an angle. 

O, a round beam, or solid cylinder. 

a = a side of the square beam ; s = a side of the triangular beam ; 
h = the perpendicular of the triangle ; c? = the diameter of the round 
beam. 

The greatest transverse strength of an equilateral triangular beam 
24* 



282 LATERAL OR TRANSVERSE STRENGTH OF BODIBB. 

vA)» expressed by no special property of (jhc figure but its area, is 



A being the area. 

But v(-j-) = ^"j ^' — i^" = ^"» and J >v^(^^ ) = i^» hfiooe, 

an equilateral triangle being constructetl that contains the given area, 
its greatest strength, in relation to the strength of any other figure 
of the same area, is ixh the dimensions /r X (^* — i-^) of that triangle. 
So, too, the least transverse strength of an equilatiTal triangular 
beam (V)» ^^^e area only of the figure l>eing known, is expressed by 

A= X Nfv(T)-WC-?^')-l^C"> 

But A' = 4/15; therefore, the triangle Ixjing constructed that con- 
tiiins the giv<*n area, its least strengtii, in relation to the strength of 
any other figure of the Siime area, may be expressed by i/i^ X 
(/* — .l>')of that triangle. 

Side^ X 1 V3 = Area of an Equilateral Triangle. 



Comparative Transverse Strengths, Sectional Areas Eipial 



O ^ AxV( A) + iV(2A)^V(A) ^ L 



0G904 



D AX V(A) 1 

D._ A X VA __ 1.4367 

O ~" A .7854 X V(A .7854) "" 1 

A _ ^(-3~)^^^-^ _ 1.62147 

D "~ Ax VA ~" 1 

D^ _ AX VA _ 1. 0682 

V'a^xVv^TV^ " ' 

A _ vC-^')>&<^> __ 1.7.S205 

'^"a^x>Jv(t):^ "" ' 

A_ vC^^-)^>fec. 2.32056 

O "■ AT854 V(A .7854) "" 1 



^ A= \rV (-3- ) , &c. 1 .34497 



4 

4 



O"" A. 7854 V (A .7864) 



LATERAL OR TRANSVERSE STRENGTH OF BODIES. 288 

Comparative Transverse Strengths, Side of Square Bar, Side of Equi- 
lateral Triangular Bar and Diameter of Round Bar, Equal. 

O _ Q- X {a-\-^n-s^ a) _ 1.0G904 __ 0.93542 

D "" a' ■*" 1 • ~" I • 

_0 _ (^-.7854)- _ 0.61685 ^^ _ 1.6212 

D "" a"^ ~ 1 ~" ~~I 

/^__ h^x{h—ks) _ 0.462019 ^^ __ 2.16441 

. D "" a^ ~" 1 "" I 

V__ hhsXVi—is) _ 0.266747 ,, _ 3.74887 

□ "" a^ ~ 1 ""1 

/^_ h-X{h — is) Jr _ 1.73205 . ^^ _ 0.57735 
V " hhsX (h—is) ~ 'hhs~ I ""1 

A __ sIrX{h — is) _ 0.749 ^^ _ 1.33511 

O "■ (d' .7854)^ ""1 "~ i 

0_ __ {d' .7854)^ _ 2.3125 ^^ _ 0.43243 

^ ~~ hlis'X{h — is) — 1 ""1 

By help of the preceding ratios it is an easy matter to find the tmns- 
verse strength of a round beam, shaft, or journal ; of a square beam, 
lying horizontal and jSiat ; of a square beam, through its diagonal ; and 
of an equilateral triangular beam, angle up or down, of any material 
of which we have an initial breaking weight, be it for a square form 
or for either of the forms enumerated. Or, if desired, we may con- 
struct a table of initial breaking weights for either of the forms enume- 
rated, having one already for either of the others. Thus, if we cube 
the diameter of a round beam in inches, multiply that cube by the 
tabular ])reaking weight of a square beam of the same material, lying 
horizontal and flat (page 279) ; multiply the result by the relative 
strength of a round beam to the square beam, the diameter of the 
former being equal to a side of the latter, and the relative strength of 
the latter being 1, and divide the last product by the length of the 
beam in feet, the quotient will be the transverse strength of the beam, 
it being fixed at one end, and the weight or load suspended from the 
other ; and if we multiply the strength thus found by 4, the quotient 
"will be the transverse strength of the beam, it being supported at both 
ends, and loaded in the middle, &c. If. we multiply the tabular or- 
initial breaking weight (page 279) of a beam of white oak, for in- 
stance, by 0.266747, the product will be the initial breaking weight 
of an equilateral triangular beam of the same material, angle down, 
or the strain tending to convex an angle ; and if we multiply the last 
initial weight by 1.73205, the p^roduct will be the initial breaking 
weight of the same beam,^ngle up, t^c. 



284 LATERAL OB TEANSTEESE STRENGTH OV BODIK 

Also, if we multiply the Tabular breaking weight, page 279, of 
cast iron, for instance, by .Gl 085, the product will be the breaking 
weif^ht of a round bar of cast iron, one foot in length and one inch 
in diameter, the bar lx?ing fixed at one end, and the weight suspended^ 
from the other ; and if we multiply the product thus obtained })y 2,* 
the result or product will be the breaking weight, or ultimate trans- 
verse strength of the same bar, under the same method of support, 
the load )>eing distributed uniformly along its whole length ; and 80 
on for Ijcams of other material and other figures. 

Example. — Required the transverse strength of an equilateral tri- 
angular beam of white oak, 8 feet long and 6 inches to a side, the 
>)eani supported at l>oth ends, and loaded in the middle, and the load 
tending to convex an angle directly. 

8 : G' :; 220 X -i X .2GG747 : G338 lbs. Ans, 

Example. — The lateral strength of a square bar of a certain ma- 
terial, of a certain length, in a certain manner supported, and in a 
ccrtuin way loade<l, is 12(ji00 ll^s. ; what is the lateral strength of a 
round bar, of the same material, same l<*ngth, in the same manner 
t^upportod, and same way loaded, the diameter of the latter being 
ts^ual to a side of the former ? 

12000 X .01G85 = 7402 lbs. Ans, 

For a Round Dcam^ Solid Cy Under y Shaft or Journal. S, /, d and 
W, VLB before — 

/:rf^::S:W 1 ^/'' : / :: W : S. 
W : S :: d' : / \ S : W :: / : </>. 

HOLLOW CYLINDERS. 

The lateral strength of a hollow cylinder is to that of a solid cyl- 
inder of the sjimc material, (quantity of matter and length, other 
things Ix'ing equal, as the greater diameter of the former is to tho 
diauK'ter of the latter ; and the lateral strength of one hollow cylin- 
der is to that of another hollow cylinder of the same material, quan- 
tity of matter and len^jth, other things being equal, as tlie greater 
diameter of the former is to the greater diameter of the latter. 

D = the greater diameter of the hollow cylinder. 

d = the diameter of a solid cylinder, of the same material and 
length as the hollow cylinder, that, alike supported and loaded, has 
the same lateral strength a* the hollow cylinder. 

;/i z= the diameter of the hollow cylinder, if converted into a solid 
cylinder, without changing it^-i length. 

o = the diameter of the bore, or interior diameter of the hollow 
cylinder. 



LATERAL OR TRANSVERSE STRENGTH OP BODIES. 285 

W, /, and S, as before. 

(P WZ 

(s^-) — ^' = i>' — ^' = <>' 

Example. — What is the transverse strength of a hollow cylmder 
of cast iron, its greater diameter being 7 inches, its interior diameter 
5 inches, and its length IG feet ; it being supported at each end, and 
loaded uniformly along its whole length ? 

684 X 8 X 7 X (7' — 5-) X 0.G1685 -M6 = 35442 lbs. Arts. 

Note. — In practice, with a view to safety, a material should not be relied on as haT- 
ing a permanent lateral strength, exceeding one-third its ultimate lateral strength 



i 



DEFLECTION OF BEAMS, SHAFTS, &c. 

w a= weight with which the beam is loaded, in pounds. 
a = tii])uliir or initial deflection, page 279. 
A = actual (Icflr'crioii in inches. 
W, C, /, /*, and I', as before. 

1 . When the beam is fired at one end, and loaded at the other ; the 
weif/ht of the ifcam not iKuifj taken into account, 

wP wPa 

Deflection varies as -7-: ; and 777-^ = A 

Ratio of Load to Brooking Woiglit, or l>cn»Mjtiun 10 Circatost Do- 
, . tvl Cbv" ,., Un^ , .^ , , , 

llcction, as -rrr— ** — 7~ "= »> • — ; i'' = nmerencc of load and 

breaking weight. 

2. Fixed at one end, and load distributed uniformly over whole 

Imgth, 

^ . . '^'^ . 3frra 

Deflection vanes as j-j^ ; and . ^ = A . 

wl 
Katio of Load to Breaking AN eight, &c., aa oTy^* 

3. Supported at both aidsy and loaded tn the middle. 

^ ^ . . "'^' , ^^^^ 

Deflection varies as 7-7 ; and ;v,^,, . = A . 

xd 
Ratio of Load to Breaking AN eight, <fec., as ,.,.. . 

4. Supported at both cnds^ and loaded uniformly along the whole 
length. 

■r. n . . '''^ , 5 waP 

Deflection vanes as 7-; ; and rr X o,..,, , = A. 
bv^ 8 32C6i;' 

wl 
Ratio of Load to Breaking AV eight, &c., as ^. ,, ^ . 

The deflection of a round beam is to that of a square beam, tne 
diameter of the former being equal to a side of the latter, and other 
things equal, as the transverse area of the srjuare l)eam to } the tran^ 

verse area of the round beam ; as ., >. .■^■- . ; as t-' nearly. 

The deflection of a hollow cylinder is to that of a solid cvHnder of 
the siime material, quantity of matter and length, other things !)eing 
equal, as the diameter of the latter to the greater diameter of the 
former; that is, their deflections vary inversely as their strengths; 
and the deflections of hollow cylinders, one with another, other 
things being equal, are inversely as their exterior diameters. 



i 



KESISTANCE TO TORSION. 

The following table shows the weight or force in pounds (mean of 
experiments), required to twist asunder bars One Inch Square and 
One Inch in Diameter of the materials named, the Weight acting 
upon the bar at One Inch from the bar's axis. 



Materials. 


Sq. Bar. 

lb.=. 


Rd. Ear 

lbs. 


Cast Steel, 


28560 


17620 


Blistered Steel, 


24380 


15040 


American Wrought Iron, 


14780 


9120 


Swedish Wrought Iron, 


13870 


8560 


Cast Iron, 


13780 


8500 


Yellow Brass, 


6860 


4230 


Cast Copper, 


6280 


3875 


Oak, 


6160 


3800 


Fir, 


6690 


4130 



American Wrought Iron 
twists and takes a per- 
manent set — square bar 
with 9640 lbs., round bar 
with 5950 lbs. applied. 
Swedish Wrought Iron 
twists and takes a per- 
manent set — square bar 
with 9790 lbs., round, 
with 6040 lbs. applied. 



C = tabular weight above, special for the case. 

W = breaking weight in pounds. 

r = leverage of applied force or W's radius of action, in inches, 
— (the perpendicular distance in inches from the axis of the shaft to 
the point on the lever or crank, where the motive power is directly 
applied.) 

c? = side of square shaft or diameter of round shaft, in inches. 





Wr. 

^- d' 


Cd?. 


Wr 

d'= • 

c 


'■ = w 



Example. — Required the weight or force necessary to twist asun- 
der a bar of square rolled iron, 2 inches to the side, the force acting 
upon the bar through a lever or crank thirty inches in length. 

14780 X 8 -r- 30 = 3941 lbs. Ans. 

Example. — What must be the diameter of a cast-iron cylinder 



288 RESISTANCE TO TORSION. 

in order that it may have a torsional strength equal to 20000 poandi 
acting upon it through a leverage of 13 feet? 

20000 X 150 -^ 8500 = ^367.00 = 7.16 inches. Ans, 

For Practical Purposes, with a view to safety, -r-p = ^' 

The above cylinder, therefore, for pnictieul purposes, other things 
remaining unchang^xl, should fiave a diameter of 

20000 X 150 X 3 -^ 8500 = >C/ 1101.18 = 10.32 inches. 

T(t find tfie ntiirJ/cr of drr/recs torsion that a given wrif/hl, tctth a 
(jivtn radius of action^ will occasion in a solid cast-iron shaft "/' "iiu u 
length and diameter, 

wlr wlr 

GOO^ ~ ' 6G0N ™ ^'' ^' ' ^ ^*°S *^® v)dght or applied 
force in p(junds ; /, the length of tlie shaft in feet ; r, w^s radius of 
action in inches ; N, the uuuiIkt of degrees torsion. 

Example. — What must bo the diameter of a solid cast-iron shaft, 
in order that if 1200 pounds force l)c applied to it, through a lever- 
age of 22 inclies, to set it in motion, the torsion shall bo but 4 de- 
grees in its entire length, its length being 20 feet? 

^ (1200 X 20 X 22 -r- GOO X -*) = 3.76 inches. An$. 
NoTR. — For rrirtic.ii lUiUs api'iicublc to Revolrlog ShafU, Joornalf or Ondfeoos, Mt 

JOUUNALS OK SlIAHS, JUigC .TJO. 

nOLLOW CYLINDERS. 

The United States Ordinance Manual furnishes the following for- 
mula, deduced by Lieutenant Kodman from the results of eiperi 
mentvS hy Major \Vado, for fmding the torsional strength of hollow 
cylinders, substituting the foregoing symbols to avoid definitions, 
namely, 

A\ = C X — n — ; 1^ being the external, and o the internal diam- 
eter of the cylinder ; W, C and r, as before. 

Of course this formula does not admit of resolution so as to find 

1) ; -;-;,- =: — r- — = </3 ; it b not mathematical, tlierefore, in 
any degree, to the transverse area of the matter in cylinders ; 

(d'- ^) X D = 0'. 

Basing our calculations upon the results of these experiments, or 
upon the foregoing formula for strength, it may bo shown that if we 



RESISTANCE TO LONGITUDINAL COMPRESSION. 



289 



take from the centre of a solid cylinder (by boring or otherwise) , a 
cylinder equal in diameter to ^ the diameter of the cylinder from 
which we take it, the hollow cylinder left will retain .987654321 — of 
the torsional strength it possessed before any of its material was ab- 
stracted ; and that if we take, in like manner, from a solid cylinder, 
a cylinder equal in diameter to i the diameter of the cylinder from 
which we take it, the cylinder thus abstracted from will still retain 
y-| of the torsional strength it possessed when solid ; also, that if we 
take from the centre of a solid cylinder J its material = ywoV its 
diameter, we shall diminish its torsional strength but 25 per cent. 



RESISTANCE TO LONGITUDINAL COMPRESSION. 

Solid bodies of the same material and length resist longitudinal 
pressure one with another, as are their transverse areas one with 
another, respectively ; but their transverse areas remaining the same, 
their power of resistance is slightly diminished by an increase of 
length ; and when the length of a body exceeds by about four to six 
times the square root of its transverse area (as a general thing for most 
substances), it cannot be cruslied by longitudinal compression ; but, 
being left free, will bend a.nd break transversely. 

The following table shows the force in pounds required to crush 
bars or blocks, in the direction of their lengths, of the materials 
named, the transverse area of each piece being one inch, and the 
length of each not more than three times nor less than onoe and a 
half the square root of its transverse area. The woods supposed to 
be seasoned, and the length of each bar of wood, in all cases, in the 
direction of its fibres longitudinally. 



^Materials. 


Povinds. 


INIaterials. 


Pounds. 


White Oak, 


3960. 


Compact Limestone, 


2270. 


White Pine, 


1600. 


Sandstone, 


1960. 


Elm, 


1280. 


Good Red Brick, 


240. 


Granite, 


3200. 


Wrought Iron,* 


116000. 


Hard Freestone, 


2770. 


Cast Iron,* 


93000. 



* Though 116000 pounds force, or thereabout, is requu-ed to crush an inch cube of 
wrought iron, and 93000 i)ounds an inch cube of cast iron, yet the former of these can bear 
but 17800 pounds, and the latter but 15300 pounds, without permanent alteration. These 
last quantities, in practice, should be taken as the ultimate strengths of these materials. 
Oak is diminished nearly one-third in length by pressure, before it crumbles j and some 
other kinds of woods are diminished full one-half. 

25 



290 



RESISTANCE TO LONGITUDINAL COMPRBSglOIf. 



C = tabular weight, foregoing. 

/ = length of column in feet. 

s = side of square column in inches. 

d = diameter of round column in inches. 

d == breadth of rectangular column in inchca. 

t = thickness of rectangular column in inches. 

W = reliable or practical resistance. 

C X transverse area = crushing weight. 
For ordinary praclical purposes. 

Solid Squaw ColumD. 

4:5^::C:W. 5« : W :: 4 : C. C : W :: 4 : j«. 

Solid liectangular Column. 

^:U::C:^y. Z»/ : W :: 4-; C. C : W :: 4 : ^/. 
Cb : W ::4 : /. O : W :: 4 : *. 

Solid Cjlindrical Column. 
5.1 :</«:: C:W. C : W :: 5.1 : rf«. 

Example. — Kc(iuircd the reliable power of a Whito Oak post to 
resist longitudinal j>reHsure, t!ic breadtli Ix'ing six inchos and the 
thickness 5 inches. 

3900 X X 3 -^ 4 = 2U700 lbs. Ans. 

Example. — Requireil the reliable or practical resistance to longi- 
tudinal compression, of an Elm shaft, G inches in diameter. 

1280 X G^ -r- 5.1 — 9U55 lbs. Ans. 

In keeping \rith the mass of matter, and the liabiUty of flcjmre re- 
movcd : any length of column, — Practical or reliable resistance. 



Wrought Iron, 
Cast Iron, 
White Oak, 



Soli.l 84)uari Column. 



17800/.^ 



W. 



4?+:t8? "" ^^ 

3900/5^ 



Solid Cjlindfieal Colanw. 






9oG2</^ 

4rf=^+.18/« 
2\lbd' 



4rf+.5/« 



= W. 



= W, 



OF CENTRES OF SURFACES fiiND CENTRES OF GRAVITY. 

SURFACES. 

To find the centre of an equilateral triangle. 
Side X .57735 =2= distance from the vertex of each angle. 
Side X .28868 = distance from the middle of each side. 

In any triangle, if a line be drawn from the vertex of either angle 
to the middle of the opposite side, it vdll pass through the centre 
of the triangle, and the centre will be on that line at | its length 
from the vertex, or \ its length from its junction with the side. It 
is at that point, therefore, where any two such lines cross each 
other. 

To find the centre of a square. 

Side X .5 == distance from the middle of each side. 

Side X .707107 = distance from the vertex of each angle. 

To find the centre of a regular pentagon. 

Side X .688188 = distance from the middle of each side. 
Side X .850644 =s distance from the vertex of each angle. 

To find the centre of a regular hexagon. 

Side X .86604 = distance from the middle of each side 
Side X 1 = distance from the vertex of each angle. 

To find the centre of a regular octagon. 

Side X 1.2071 = distance from the middle of each side. 
Side X 1.3065 = distance from the vertex of each angle. 

To find the centre of a rectangle, 

J V (^^ + «') J <^r V (i A)2 -\- (4 (z)2 = distance from the vertex 
of each angle. 

J the length of A from the middle of g, and 
J the length of a from the middle of A, A being one of the 
longer sides, and a one of the shorter. 

To find the centre of a rhombus. 

Side X .5 = distance from the middle of each side. 
. From the angles ; on a diagonal at half its length. 

To find the centre of a rhomboid. 
From the middle of A, distant ^ a, and from the middle of a, dis- 



292 CENTRES OF SURFACES. 

tant i A, to the same point ; A being one of the longer sides, and a 
one of the shorter. From the angles, same as in the rhombus. 

To find the centre of a trapezoid, 

b A + 2rt 

o X "7 — i = distance from the middle of A, on the line b ; A 

being the longer of the two parallel sides, a the shorter, and b a line 
passing from the middle of A to the middle of a. 

To find the centre of a trapezium. 

Draw a diagonal and find the centre of each of the triangles 
thereby formed, and pass a line from one centre to the other. Thon 
draw tlic otiior diagonal, and find the centre of each of the trianirl' s 
th<;roby formed, and connect those centres t<>gether by | • ' •■ - 
line from one to the other. The point at which the lines < 
other is the centre of the figure. And this principle is a^-l-nLaMiu 
to all (|uadrilatorals. 

lu a circle — 

Circumference X .159155 = radius. 

To find the centre of a semicircle, 

Radius 

2T^> = distance on bisecting radius from the middle of the 

chord or centre of the circle. 

/ /radius\2\ 

V( Iiadius'--|- ( « ^ ) ) "^ distance from both extremities of 

the arc ; n being the ratio of circumference to diameter, that is, n 
= 3.141G. 

Kadi us X .42441 =3 distance on bisecting radius from the middle 

of the chord. 

Radius -T- 1.737354, or nvdius X .575588 =« distance on bisecting 
radius from the vertex. 

To find the centre of the segment oj n nrdc. 

Chord of the segment'^ 

TO v. — — ^ — »• „^ . ... » = distance from the centre of the circle, 
11: X iirea or segment ' 

on the versed sine. 

V (4 chord of segment^ -|- distance of centre of segment from 
middle of chord-) = distance of centre from both extremities of the 
arc. 

Radius of circle — distance from centre of circle = distivncc from 
vertex. 



CENTBES OP GRAVITY. 293 

To find the centre of a sector of a circle, 

I chord of arc X radius of circle 

\ n — c = distance from centre of circle, 

length ot arc ' 

on radius perpendicular to the chord, and bisecting the sector. 

/s/ (i chord of arc2 -|- (radius — versed sine — distance from 
centre of circle)-) = distance from both extremities of the arc. 

Radius — distance from Centre of circle = distance from vertex. 

To find tlie centre of a parabola. 

On its axis at f its length from the middle of the base. 

V (4 base^ -|- distance from base-) = distance from 'both extrem- 
ities of the curve. 

Axis -r- distance from base ^ distance from vertex. 

In an Elli^pse, the centre is at a right section of the figure, as in 
the parallelogram. 

In a Zone, the centre is at a right section of the figure, as in a 
trapezium. 



SOLIDS. 

The centre of gravity of a Cube is its geometrical centre, and the 
same is true for a Right Prism, Parallelopiped, Cylinder, Sphere, 
Spheroid, Ellipsoid, and any Spindle. It is also true for the Middle 
Frustum of any spindle, the Middle Frustum of a spheroid, the two 
F^ual Frustums of a j)araboloid, and the two Equal Frustums of a 
cone. 

In a Cone and Pyramid, the centre of gravity is in the axis, at | 
the length of the axis from the base. 

In a Paraboloid, the centre of gravity is in the axis, at J the 
length of the axis from the base. 

To find the centre of gravity of a frustum of a cone or frustum of a 
regular pyramid. 

(D + ^)2 -f 2 D2 height 

7d 1 d\- D / ^ — 4 — ^^ distance on the axis from the centre 

of the less base ; D being the diameter of the greater base, and d 
the diameter of the less, in a frustum of a cone ; or, D being a 
side of the greater base, and d a side of the less, in a frustum of a 
regular pyramid. 

25* 



294 CENTRES OF OSCILLATION AND I'EIiCUSSION. 

To find the centre of gravity of a prismoid, 

(VA + Va)- + 2A height 

7-7-7 — ■. J—-, 7-7-7:^ — TT X — T— = distance on the j 

( V A + Va)- — VAX Va ** 

from the less base ; A being the area of the greater base, and a the 
area of the less. And this rule is applicable to the frustum of a 
pyramid, of any number of equal sides, and to the frustum of a 
eone. 

To find the centre of gravity of a frustum of a paraboloid. 

D2X2 + /-^ height 

— |v.j . .J - ^ — « — »« distance on the axis from the loss baae ; 

D ])eiug the diameter of the greater base, and d the diameter of the 

less. 

To find the centre of gravity of a spherical segment, 

3.1416 X >Hiyht of segment^ X (rmliuK of gphcre — j height of •egineot)> __ 
soUd ooDlcoU of legmeiit 
iancc from the centre of the sphere, on the axis of the le gmeui \ the toHd o mtUm i m Mad 
lines Tx'ing in Uie sanie denominatioo of measare. 

Iladius of sphere — distance from centre of sphere to centre of 
gravity of segment = distance on the axis of the segment from the 
segment's vertex. 

Uiullus of ba»e of sefnuent^ -4- height of sefnnent' „ . ^ 

N..Tr _ i—-;^ — ■ ; = rttdioB of iphcre. 

height of segment X 2 

To find the centre of gravity of a spherical srrtor. 

(Ra<lius — ^ v.r3o«l sine) X 3 ,, ...... . . 

^ ^ distance on biBecUng radiuB fK>m centre of sphere. 

4 

To find the centre of gravity of a system of bodies, 

oXt+«'Xt- + «''Xft",&c.^^^^ ^„^ 1 a, a-, a", *c, b<to, a» «M 
a ~\- a ~j~ a \ «c. 
contents or weiKht^i, and ^, b., b'\ &c., Uic distances of their respectire oentrea of gra f My 
from the given plane. 



OF CENTRES OF OSCILLATION AND PEliCUbSiUN. 

The centre of oscillation applies to bodies fixed at one end and 
vibrating in space ; and the centre of percussion applies to bodies 
revolving aroun'l a fixed axis. The two centres, in the same body, 
are at the same point. 






. CENTRES OP OSCILLATION AND PERCUSSION. 295 

The centre of oscillation or percussion is that point in a body, 
under one or the other of the supposed motions, in which the whole 
motion, tendency to motion and force in the body, may be supposed 
collected. It is that point, therefore, in the body under motion, 
that would strike any obstacle with the greatest effect, or, if met by 
a staying force, would serve to stop the motion and tendency to mo 
tion, of the whole mass, at the same instant. 

In Pendulums, or rods of uniform diameter and density, having 
one end fixed, and a ball or weight appended to the other : 

Weight of rod X h length of rod = momentum of rod. 

Weight of ball X (length of rod ~\- radius of ball) = momentum of ball. 

Weight of rod X lengtli of rod^ 
— ■ :^= force of rod. 

Weight of ball X (length of rod -\- radius of ball)^ = force of ball. 

Force of rod 4- force of ball ,. „ , . , 
:: — r-7 :: — r: = distance from the pomt of suspension (x 

momentum of rod -|- momentum of ball 

axis of motion, to the centre of oscillation or percussion. 

And in a rod of uniform diameter and density, having one end 
jBxed, and two or more balls or weights attached : 

Weight of rod X =} length of rod = momentum of rod. 

Weight of 1st ball X distance from axis of motion = momentum of 1st ball. 

Weight of 2d ball X distance from axis of motion = momentimi of 2d ball. 

Weight of rod X length of rod^ . . , 
= force of rod. 

Weight of 1st ball x distance from axis of motion^ = force of 1st baJL 
Weight of 2d ball X distance from axis of motion^ = force of 2d ball. 

Force of rod 4- sum of the forces of the balls , „ , . - . 
■ = distance from the axia of motion to 

momentum of rod -\- momenta of the balls 

the centre of oscillation or percussion. 

By another Rule. — Suspend the body freely by a fixed point, and 
cause it to vibrate in small arcs, and note the number of vibrations 
it makes per minute. Then — 

Number of vibrations per minute^ : 602 : : length of pendulum 
that vibrates seconds in the respective locality : distance from point 
of suspension to centre of oscillation. 

Example. — The rod is 10 feet in length, and weighs 21 pounds ; 
the weight of a ball, having its centre at the lower extremity of the 
rod, is 16 pounds, and that of another, afl&xed to the rod 4 feet from 
the point of suspension, is 9 pounds ; required the centre of oscilla- 
tion in the system. 



296 CENTRES OP OSCILLATION AND PEBCTJS8I0N. * 

i(21 X 10-) + (9 X 4^) + (IG X 102) 

21 X 5 + (9X4) + (10X10) ^ ^-^2 ^^^ ^""^"^ ^^« 
point of suspension. Aiis, ■ 

Example. — If, on suspending the body supposed in the last ex- 1 
ample by its unloaded end, it is found to vibrato 38 times in u min- 
ute, whjit is the distance from the point of suspensiDn to tho. centre 
of oscillation ? 

382 : G02 : : 3.2584G : 8.12 feet. Atis. 

In a Rii^IU Line, Cylinder or R/uiialrral Redan fpilar Prism, one 

end bein^ in the axis of motion, the centre of oscillation or percu»- j 

Bion is distant from that end J the length of the line, cylinder or f 

prism. f 

Tiic axis of motion being in the vertex of the fi^x^re, and the fieure | 
inovingy7a/trj5e, the centre of oscillation or percussion is distant from I 
the vertex. 

In an isosceles triangle, § its height ; 
In a circle, J its radius ; 
In a parabola, ^ its height. 

But the axis of motion being in the vertex of the figure, and the 
fip^uro moving sirlewise, the centre of oscillation or percussion is dis- 
tant from the vertex, 

In a circle, J its diameter ; 

3 arc X radius 
In a sector of a circle, « . » — ; 

In a parabola, ^ height + J ]>animctcr ; 

radius of base'-' 
In a cone, i axis + t;-^^ ; 

radius- X 2 
^" " «P^^^^^» radius X 5" + '^^*"' ^ 
In a rectangle, suspended by one angle, J the diagonal ; 
In a parabola, suspended by the middle of its base, ^ axis + i 
parameter ; 

radius^ X 2 
In a sphere, suspended by a thread , 5 ^ (radius + length of thread) 
+ radius + length of thread. 

Example. — Required the length of a rod of uniform diameter 
and density, that, being suspended by one end, will vibrato secondfl 
in the latitude of New York. 

2:3:: 39.10153 : 58.6523 inches. Ans, 



CENTRE OP OSCILLATION AND PERCUSSION. 297 

Example. — A ball of 4 inches radius is suspended by a string 24 
inches in length ; required the centre of oscillation or percussion in 
the system. 

4' X 2 32 

(4 .-I, 24) xT "^ 140 "^ -^^^^ + 4 + 24 = 28.2286 inches from 
the point of suspension. Ans. 

Example. — Required the centre of oscillation in a parabola whose 
height is 10 inches and base 8 inches, supposing the parabola to be 
vibrating sidewise. 



10 X 5 4' 



10X3 



= 7.676 inches from vertex. Ans. 



The centre of oscillation in a sphere suspended by a point in its circum- 
ference being given ^ to find the radius of the sphere. 

h — "y = radius ; h being the distance from the point of suspen- 
sion to centre of oscillation. 

Example. — Required the radius of a sphere, that, being suspended 
by a point in its surface, will vibrate seconds at 23"^ of latitude. 

39 01206 — ^M1?21>1^ = 27.865 + inches. Ans. 

The radius of the ball or bob of a pendulum being given ^ to find the 
length of the rod or string by which that ball must be suspended, in 
order that the pendulum may vibrate seconds at a given locality. 

/r2 X 2 \ 
P — I p v^ - ) -)- ?'= ^ y P being the length of a pendulum that 

vibrates seconds in the respective locality, r the radius of the ball, 
and I the length of the string or rod ; the last supposed to be with- 
out weight. 

Example. — The radius of a ball being 4 inches, required the 
length of a thread, (supposed without weight,) whereby to form a 
pendulum with that ball, that will vibrate seconds in the latitude 
of New York. 

39.10153 — ( — 1-21_? f- 4^ = 34.937 inches. Ans. 

\39.10153 X 5 ' / 



298 CENTRE OP GYRATION. 

CENTRE OF GYRATION. 

The centre of gyration is that point in a revolving body or sptem 
of bodies, in which, if the whole quantity of matter were collected, 
the angular velocity would be the same ; that is, the momentum or 
quantity of motion in the revolving mass is centred at this point. 

The centre of gyration is at a greater distance from the axis of 
motion than the centre of gravity, and at a less distance from that 
axis than the centre of percussion. Ite distance from that axis is a 
mean proportional between the two, in the same mass. 

To find the centre of gyration in a body or system of bodies. 

Rule 1. — Multinly the distance of the centre of percussion from 
the axis of motion, l)y the distance of the centre of gmvity from the 
axis of motion, and the 8r|uare root of the product will bo the dis- 
tance of tlio centre of gjnition from the axis of motion. 

Rule 2. — Multi|.Iy the weight of the several ]>article8 by the 
squares of their distunces from the axis of motion, and divide the 
sum of the pnxhicts by the sum of the weights ; the square root of 
the quotient will Ix; the distiince of the centre of g^-ration from the 
axis of motion. 

Example. — On a lever (8up|x>sod without weight) revolving 
about one end, there are placed — one weiglit of 4 lbs., at .'J feet 
from the axis of motion ; one of ll>s., at 4 feet ; and one of 5 lbs., 
at 5 feet from that jixis ; recjuircd the centre of gyration in the 
system. 

^//4X3^+JGX4») -f (5 X ^)\ 

^ \ 4^6 jr 5 ) * 4-14 feet from the axis 

of motion. Ans, 

A weight of 15 lbs., therefore, placed 4.14 feet from the axis of 
motion, and revolving in the same time, would have the same im- 
petus or momentum as the three weights in their respective places. 

Example. — A lever G feet in length, and weighing 14 lbs., is 
revolving about one end ; at 4 feet from the axis o? motion, on the 
lever, is placed a weight of 5 ll>8., and at 6 feet from that axis, 
another of 8 lbs. ; required the centre of gyration in the system. 

axis of motion. Arts, 

Example. — Required the centre of gravity and centre of percus- 
eion in the last mentioned system. 



CENTEE 01 »TEA^XON. 299 

14X&+(5X4) + (8X6) , r ; A 
1 4 4, 5 4- 8 = 4.074, centre of gravity. Arts, 

14Xj^+(5X4^+(8X62) .n^o ^ c A 

^ — — ^— ^ ^^ =4.873, centre of percus. Ans, 

14 X f + (5 X 4) + (8 X 6) 

Example. — The centre of gravity in a body, or system of bodies, 
being 4 feet from the axis of motion, and the centre of percussion 6 
feet ; at what distance from that axis is the centre of gyration ? 

V (4 X 6) = 4.899 feet. Ans. 

The centre of gyration is distant from the aods of motion : — 

In a straight, uniform rod, revolving about one end ; the length 

V /length of rod2\ 
,, , .. ( — ^"3 ). 

■ In a circular plate, revolving on its centre, or a cylinder, revolv- 
* ing about its axis, or a v^heel of uniform thickness, revolving about 
. its axis ; the radius X .7071. 

In a circular plate, revolving about one of its diameters as an 
( axis ; the radius X '5. 

In a thin, hollow sphere, revolving about one of its diameters aa 
an axis ; the radius X .8164. 

In a cone, revolving about its axis ; the radius of the base X 
.5477. 

In a four-sided equilateral pyramid, revolving about its apex ; 
the axis X -866. 

In a paraboloid, revolving about its axis ; the radius of the base 
X .57735. 

In a sphere, revolving about one of its diameters as an axis ; the 
radius X .6325. 

In a straight lever, the arms being R and r, = w /^ r^ V 3 

In wheels in general revolving about their axes, = 
,/R X ^2 X 2 + A X ^' X 2\ 

* VI (R-4- A^ X 2 / ' "^ being the weight of the 

rim, r the radius of the wheel, I the length of the arms, and A the 
weight of the arms. 

In a water-wheel in operation, = 
,(1^ X r^ X 2 + A X ^^ X 2 + W X r2 ^ , ^,. 

V i^^ (R + A-f W)X2 > ^^^^^y ' ^ ^^^g 

the weight of the water on loaded arch. 



300 CINTBAL FORCB. 

The En-erg y of a revolving body = 
Velrxjity of centre of gyration in feet per second^ 
'-^^ X weight of 

revolving body. 

The energy of a revolving body is its weight multiplied into the 
height due to the vchjcity with which the centre of g>Tiition movet^ 
in its circle. The height in feet, due to a velocity io feet per seoood 
= (velocity -r- 8.02)% or velocity^ ^ G4J. 

ExAMPLK. — A grindstone two feet in diameter and weighing IW 
lbs., makes 80 revolutions per minute; required the energy with 
which the 8t(^)nc moves, or the mechanical power that must bo cum* 
muuicuted, to give it that motion. 

/2X .7071 X 3.1410 X ^OV I 

y jjQ • — ) -r- 641 = .M5C, energy, the wei|^. 

being 1, and .5450 X 140 = 70.38 Iba. Aks. 



CENTRAL FORCES. ■ 

The Central Forcxs are the centrifugal force and centripetal force: 
these forces are opposed to each other. Iho fonner is the f i 

which a bixly, moving in a curve, tends to fly off from the a.\ : iii 
motion, and the latter is the force that maintains the body in iti 

curvilinear path. 

To find the centrifugal force of a body. 

RovolatioiM p» r mlrmt<r* X dtarocter of centre of iryration*! cirde In fc<t 

6806 ' 

K»l force, Uie weight being 1. 

/Vi Kvity of rent re of ^ryration tn feet per MOondNS 

V j^ =-- ; -rtwkxthedlrtMoeorthii 

of gyration from centre of motion s= oentrifiigal force, the weight being 1. 

If wo let 

velocity of centre of g\'ration in feet \ 

centre ofgymtioilj^ 



V represent velocity of centre of g\'ration in feet per second, 
r ** nulius, in feet, of circle described bye 



w ** weight of body, 

c " centrifugal force ; then 



t-- X »^^ 
rX32i 

rX32^ 



c X 32^ X r 



- 6_^^W 



v(^'i^') 



FLY-WHEELS. — THE GOVERNOR. 801 

Example. — Required the centrifugal force of a fly-wheel whose 
jentre of gyration is 8 feet from the axis of motion, and which centre 
moves with a velocity of 34 feet per second. 

( -^ J H- (8 X 2) = 4.49 times the weight of the wheel. Ans. 



In fly-wheels in general, 

Diameter of wheel in feet X .6136 
time in seconds of one revolutions 



X weight of rim = centrifugal force, nearly. 



FLY-WHEELS. 

Fly-wheels are used both as regulators of force and magazines of 
power; when, more especially for the Tormer purpose, they should 
be placed as near as practicable to the prime mover; and when, 
more especially for the latter, as near as practicable to the working 
point. 

To find the weight of the rim of a fly-wheel j>roperfor an engine of a 
given horse-power. 

Horse-power of engine X 136S , . . , . . . ,,^ 

— — , — — — ; — : : = requisite weight of rim in 100 

diameter of wheel in feet X revolutions per mmute 

pounds. 

Ex^vMPLE. — What should be the weight of the rim of the fly-wheel 
of a 40 horse-power engine, the wheel being 16 feet in diameter, and 
making 42 revolutions per minute ? 

40 X 1368 



THE GOVERNOR. 

The Goterxor is a regulator of force merely, and acts upon the 
principle of central forces. It makes half as many revolutions in 
any given time, as a pendulum, the length of which is equal to the 
perpendicular distance between the point at which the arms are 
suspended and the plane in which the centre of oscillation in the 
system moves, makes in the same time. 

If we let 

I = perpendicular distance referred to, in inches, 

r = number of revolutions made by the system per minute, 

Then, 

/I87.6y_ ^^ _, 187.6. 



and 



26 \ r J ' s/l 



S02 FORCE OP GRAVITY. 

Example. — A goveraor makes 40 revolutions a minuto ; what is 
the perpendicular distance from the point at which the arms aro 
suspended, to the plane in which the centre of oscillation moves? 

187.6 -^- 40 = 4.09 X 4. GO = 22 inches. Ans, 
\/l X -31980 = time in seconds of each revolution. 



FORCE OF GRAVITY. 

Tire Force of Gravity is greatest at the earth's surface, and de- 
croiises from the surface to the centre as the disUmce from the sur- 
face increases ; from the surface upwurtls it decreases as the square 
of the distance from the centre, in semi-diametors of tlio earth. 

The semi-diametor is usually tiken at 4000 miles. 

A hodj, therefore, wcigliing at the earth's surface 600 lbs., 
would, were it removed lOOO miles from the surface towards the 
the centre, weigh 

4000 : 500 : : 3000 : 375 lbs. 

And were the same body placed 1000 miles above the earth's sur- 
face, it would ho 

(4000 + 1000) -5- 4000 =» li semi-diameters from 'the centre, and 
At that locality would weigli. — 

500-4- l.2r)- = oi:ulbs. 

And in order that the same body may weigh but 250 lbs., it must 
be elevated above the surface 

V(500 -r- 230) = 1.4142 scmi-dianictors from the centre, or 
(4000 X 1.4142)— 4000 = 1057 mUes. 

NdTK. — Tlio force of pravlty at any locality, aa meaaMrcd by the descent of a body 
fallir)i; frtfly throiiirh space, is e(|ual in feet per second to the length of a peodoliun in 
fi>ot that vU^rates seconds at Uuit locility X -^-O^iSa, for the first second of the body*s de- 
scent. 

Thus, the force of gravity of a body for the first second of its de- 
scent, at the level of the sea, at the "latitude of 41°, is 3.25846 X 
4.93483 = 1G.08 feet. 

To find the time which a body will be iri falling from rest, through any 
space, the space fallen through being given, or the majcimum velocity 
attained in falling being given, 

V /Space fiiUen through in feet\ . , .., 

I ) = time of fiilling In seconds. 



FOROE OF GRAVITY. 303 

Maximum velocity attained in feet per second ^. ^ , ,^. . 
= tane of laUing in seconds. 

Space fallen through in feet X 2 

- = tame (rf feJling in seconds. 



maximum velocity attained in feet per second 



To find the maximum velocity attained by a body falling freely from restj 
through space, the space fallen through being given, or the time of de- 
scending being given, • 

\/ (Space fallen through in feet X ^^~i^) X 2 = maximum ve- 
locity attained in feet per second. 

Time of falling in seconds X 32|^ = maximum velocity attained in 
feet per second. 

Space fallen through in feet V 2 . 

: -— — — : , = maximum yelocity attamed in feet per second. 

tmie of faUing m seconds 

To find the space fallen through by a body falling freely from rest, the 
time of descending being given, or the maximum velocity attained in 
falling being given. 

Time of falling in seconds^ X l^xV = space fallen through in feet. 

Maximum velocity attained in feet per second^ 



64^ 



: space fallen through in feet. 



Maximum velocity attained in feet per second X 4 time of falling 
In seconds = space fallen through in feet. 

To find the height of a stream projected vertically from a pipe. 

Quantity in cul). ft. discliarged per minute X 2.4 . , . . - . 

r. — \ : : r— : — \ == maximum velocity in feet per 

area ot dischargmg orilice in mches 

second, or velocity in feet per second when the stream leaves the pipe. 

/Maximum velocity of stream in feet per seconds^ , . , . ^ . , 
\^ — — ^^ ) = height m feet due to that ve- 
locity, (M* height in feet to which the stream will ascend. 

Example. — From a fire-engine there was discharged through a 
pipe I of an inch in diameter, 15 cubic feet of water in 1 minute, the 
pipe being directed vertically ; what height did the stream attain? 



(: 



^^ X 1728 ) = 103.22 feet, Ans. 

.75 X .75 X .7854 X 12 X 60 X 8.02/ 



To find the force or power requisite to produce the foregoing result. 

Height of column of water equal to pressure of atmosphere (33.87 feet) 



pressure of atmosphere on 1 square inch of surface (ll.T lbs.) 
oolumu of water of 1 in:h transverse area, weightug 1 lb, = 2.305 feet } then, 



height of 



L 



304 PENDULUMS. 

Pounds of water dlscharj?e<l per B«x)nd X 2.305 , ...-_, 
^ mATtmnm Telooity In feet per fei* 

area of discharging orifice in inches 

ond, and 

Maximum velocity in feet per second X weight in lbs. discharged 
per second = momentum in lbs., or force in lbs. required. 

V (Height of projection in feet X IGyW) X 2 ^ maximum veloc- 
ity in feet per second. 

Example. — Required the constant force requisite to project a con- 
tinuous stream of water perpendicularly to the heignt of 103.22 
feet, through a pipe J of an inch in diameter. 

>v/(103.22X16t^7)X2 
2.305 -r .75 -^ X l^CA X V (103.22 X 10^) X 2 = 1275 Ibe. Ans. 

15 X C2.5 X 2.305 15 X <^2.5 
^^' GO X .75* X .7854 ^ GO =" ^-' ' ^*'^- 

N»iTK. — Tlio f r. r i'<ct a body rerticany to r ^ eqaal to 

the force or nii»ni' i.U body will rctom to t . noc pro- 

jected. The time < ling is Cf)ual tn thai oocw; 



OF PENDULUMS. 

Pendulums of the same length vibrate faster the farther they aro 
removed from the equator, cither north or south. 

The length of a pendulum to vii>ratc seconds, or 60 times in a 
minute, at the level of the sea, at the temperature of 00^ F., at the 
latitude of Trinidad, is 39.O1S70 inches; at Jamaica, 39.03508 ; at 
New York, 39.10153; at Bordeaux, 31). 11 2S2; at Paris, 39.12843 ; 
at London, 39.1393; at Edin))urgh, 39.1554; and at Greenland, 
39.20328 inches. 

To find the length of a pcndii/um that will make a given number of 
i^ibrat'wns in a given time. 

Rule. — As the numl)er of vibrations required in one minute is to 

60 vibrations, so is tlio square root of the length of a pendulum that 
makes GO vibratic^ns in a minute at the respective locality, to the 
square root of the length of the pendulum required. 

Example. — Required the length of a pendulum that will vibrate 
half seconds at the level of the sea, at the latitude of New York. 

120 : GO : : V39. 10153 : 3.12G5G1, and 
3.12G561 X 3.12G5G1 = 9.77538 inches. Ans. 



SCREW-CUTTING IN A LATHE, 305 

Td find the number of vibrations that a pendulum of a given length will 
make in a given time. 

Rule. — As the square root of the length of the given pendulum 
is to the square root of the length of a pendulum that vibrates sec- 
onds in the respective locality, so is 60 vibrations to the number of 
vibrations it will make per minute. 

Example. . — Required the number of vibrations that a pendulum 
24 inches in length will make per minute, at the level of the sea, at 
the latitude of New York. 

V24 : V39. 10153 : : 60 : 76.585. Ans, 

To find the length of a pendulum that will vibrate as many times in a 
minute as there are inches in that pendulum'' s length. 

Rule. — Multiply the square root of the length of a pendulum 
that vibrates seconds in the respective locality, by 60, find the cube 
root of the product, and the square of the last root will be the length 
sought. 

Example. — What must be the length of a pendulum, in order that 
its length in inches and the number of its vibrations per minute, at 
the latitude of New York, may be equal? 

V (39.10153) X 60 = 375.1873, and 
/iy375.1873 = 7.2125, and 
7.2125=2 = 52.02 inches. Ans. 

Note. —The length of a pendulum is the distance from the point of saspenaon to tiM 
centre of oscillation. See Centees of Oscillation jlnd P£acuss£ON. 



SCREW-CUTTING IN A LATHE. 

Screws to any degree of fineness not exceeding about I inch pitch, 
that is, having not more than about 8 threads in an inch, may be cut in 
a lathe by means of a wheel on the end of the leading screw, and an- 
other on the end of the lathe spindle, with a carrier, or intermediate 
wheel, called a stud-wheel^ to connect them. And all that is neces- 
sary in this kind of training, in order to obtain a screw of any given 
or predetermined pit^h, is, that the wheel on the end of the leading 
screw bear the same ratio to that on the end of the lathe spindle, 
that the pitch of the screw employed bears to that of the screw 
intended. i 

The carrier, whatever be the number of its teeth, and the same 
would be true if two or mure caiTiers, of whatever size, were employed, 
26* 



306 SCMW-CUTTTXO IN A LATITE, 

in no way affects the ratio of velocity between the wheels alloded 
to. It receives the motion directly from one of them, and convey» 
it directly to the other, and is introduced to obviate the necessity of 
employing larger wheels. It is both a driven and fir'rvcr in the train, 
and, consequently, in calculating the gearing to produce a screw of 
a given or required pitch, needs not to be taken into account. 

ijut screws of a greater dcCTce of fineness than about 8 threads ia 
an inch, more especially if the pitch of the screw attached to tho 
lathe be large, are more conveniently cut by helf> of two intcnnodi- 
ate wheels, of unequal sizes, both u|>on tlie same shaft, called a stud- 
wheel and pinion ; whereby the requisite vehx^ity of the »pindle may 
be more effectively obtained, and with a nearer approach to uni- 
formity in size of the wheels employed. When this last uuxle of 
training is resorted to, all tlie wheels in the tniin are influential in 
detenuinin^ tho pitch the contemplated screw will take ; and ihB 
equation will l>e as the product of the drivers to the product of tho 
driven ; that is — 

K we let a = pitch of leading screw, ^ 

a' ^ No. of teeth in stud pinion, ydriverSf 

d' =a No. of teeth in spindle-wheel, > 

h = pitch of con tempt a toil screw, \ 

y = No. of teeth in stud-wheel, Wriroi, 

y ss No. of teeth in leading screw-wheel, ) 

We shall have, ^ X a' X a" = ^ X // X ^". 

And, in the train first supposed, whether with or withoat car- 
tiers, — 

.flX^' = *X^'; 

And the product of either full side of the equation, divided by the 
product of either two terms in the opposite, will give the remaining 

term, or term souglit. 

Ex.\iiPLE. — a is i inch, d* has 24 teeth, and ^ is to equal \ inch ; 
required the requisite numl:»er of teeth in h'\ 

j X 24 = 9, aiKi 9 -^ 1 = 3G teeUi. Am, 
Or, (.375 X 24) ^ .25 = 3G ; or, .25 : IM : : .375 : 36. Ans, 

Example. — d' has 60 teeth, a' has 20, and a = i inch ; h" has 120 * 
teeth, and h is fixed upon at -i\ inch ; required the number of teeth 

requisite for h' , 

(i X 20 X 60) -^ (120 X tV) = ^^ teeth in ¥, Ans, 



Or, .5 X 20 X 60 ~ 120 X .0025 = 80. Ans. 



SCREW-CUTTING IN A LATHE, 307 

But it is more convenient, frequently, in solving problems of this 
nature, to substitute the number of threads in a given length of the 
screws, for their pitches, respectively. 

Thus, if we let a = No. of threads in an inch of leading screw, 

b = No. of threads in an in. of contemplated screw, 

We shall have, as an equation for the foregoing train — 

That is, the product of the teeth in the driving-wheels multiplied 
by the number of threads in an inch of the contemplated screw, is 
equal to the product of the teeth in the driven wheels multiplied by 
the number of threads iu an inch of the leading screw. 

Example. — «' has 20 teeth, n" has GO, and b is to have IG threads 
in an inch ; a has 2 threads hi an inch, and b'^ has 120 teeth ; re- 
quired the number of teeth requisite for b'. 

20 X GO X IG 

1^0 y ^ — A715. 

ExAifPLE. — The leading screw has 2 threads in an inch, the lead- 
ing screw-wheel has 120 teeth, the stud-wheel 80 teeth, and the 
spindle- wheel 60 teeth ; required the number of teeth that must be 
in the^tud pinion, in order that the contemplated screw may have 
IG threads in an inch. 

t 80 

^X ^^ X 80 = "4" = 20teethma'. Am. 

00 X U 



The following table exhibits the change wheels proper for cutting 
screws of various pitches from 1 inch to -j^g- inch, or from one thread 
in an inch to 3G threads in an inch, the leading screw having a pitch 
of i inch. 

Note. If the Icadinji^ screw havt- 3 threads iu an inch, the follow^ing table is made ap- 
plicable by changing either of the driven wheels for one having ^ less number of teeth. 
3 : 2 : : tabular driven : driven required, when the leading screw has \ inch pitch, and 4 : 
2 : : tabular driven : driven required, when the leiiding screw has i inch pitch, &c. Or 
2 : 3 : : either of the tabular drivers : driver required in its stead, when the leading screw 
has 3 threads in an inch, and 2 : 4 : : either of the tabular drivers : driver required in it3 
stead, when the leading screw has 4 threads in an inch, &c. 

Pitch of leading screw X No. teeth in stud-wheel X No. teeth in screw-wheel = pitch of 
contemplated screw X No. teeth in stud-pinion X No. teeth in spindle-wheel, in all 
Instances. 

The tabular wheels may be taken in any proportion, greater or less. 



II 



808 STKAM AND THE STKAM MtfOnnL 

m 

TABLE 

Cff Change Whc/jls far Scrcw-cui/ing ; the tradtng hereto being of | 
inch pilch ^ or ronUiining 2 threads in an inch. 



tl 

is 


N\2mt>cr of 
tec'lh in 


.3 

11 

»ff a. 


Number of troth in 


il 

i ^ 

^*5 


Number of teeth In 


1 

1 


1 
if 

1 


1 

1 
'2. 

1 


1 


1 

5 


1 


1 
1- 


1 


1 

2 


.J 


1 


80 


40 


»l 


40 


55 


20 


60 


r.) 


50 


1)5 


20 


100 


Vi 


80 


5<) 


^i 


«K) 


HTj 


20 


'.HI 


11)4 


80 


120 


20 


130 


li 


80 


00 


^ 


00 


70 


20 


75 


20 


60 


100 


20 


120 


n 


80 


70 


•>4 


IK) 


90 


20 


1)5 


201 


40 


IK) 


20 


«K) 


♦2 


80 


80 


^•»^ 


40 


60 


20 


65 


21 


80 


120 


20 


140 


*^l 


HO 


'M) 


10 


(iO 


75 


20 


80 


22 


60 


no 


20 


120 


'-^i 


80 


10(1 


104 


60 


70 


20 


75 


224 


80 


120 


20 


150 


'^\ 


80 


no 


11 


60 


55 


20 


120 


22] 


80 


130 


20 


140 


3 


80 


120 


12 


1)0 


•H) 


20 


120 


2;;] 


40 


'.•5 


20 


1<M) 


^ 


80 


180 


12^ 


60 


85 


20. 


IM) 


24 


65 


120 


20 


i;;o 


H 


80 


MO 


13 


IH) 


•K) 


20 


liiO 


25 


60 


100 


20 


l.V) 


H 


80 


i:>o 


134 


60 


IH) 


20 


IK) 


254 


30 


8;3 


20 


IK) 


4 


40 


80 


13^ 


80 


100 


20 


no 


26 


70 


180 


20 


no 


41 


40 


Ki 


14 


IK) 


IK) 


20 


140 


27 


40 


'M) 


20 


120 


n 


40 


•10 


141 


60 


IK) 


20 


1)5 


274 


40 


100 


20 


no 


H 


40 


\)'^ 


15 


1)0 


IM) 


20 


150 


28 


75 


140 


20 


150 


5 


40 


llHI 


ir, 


60 


80 


20 


120 


2S4 


30 


IM) 


20 


95 


6i 


40 


no 


16i 


80 


100 


20 


130 


30 


70 


140 


20 


150 


r> 


40 


120 


1^4 


80 


no 


20 


120 


32 


30 


80 


20 


1-20 


64 


40 


mo 


17 


45 


86 


20 


IK) 


33 


40 


110 


20 


120 


7 


40 


140 


174 


80 


100 


20 


140 


84 


80 


85 


20 


120 


74 


40 


150 


18 


40 


60 


20 


120 


35 


60 


140 


20 


150 


8 


30 


120 


183 


80 


100 


20 


150 


36 


30 


90 


20 


120 



OF STEAM AND TlIE STEAM ENGINE. 

It has been found by experiment that the force rc«|ui8itc to over- 
come the friction of a locomotive cnrrinc and attendant maohinorv, 
the enj^ine being without load, is equal to a pressure of alx^ut 1.15 
ibs. cflbctivc on each square inch of the cjlinder's cross sectional 



STEAM AND THE STEAM ENGINE, SW 

area, or equal to 1 lb. effective for the engine, and 0.15 lb, effective 
for the attendant machinery ; this item, however, for practical pur- 
poses, is usually taken at 1^ lbs. ; by effective pressure is meant a 
pressure over and above that of the atmosphere, or over and above 
14.7 lbs. on each square iiich of surface. 

If we let 

d = diameter of cylinder in inches, 

5 = stroke of piston in feet, 

r = revolutions per minute, 

p = mean effective pressure in \]ys. per square inch, as shown by 
the indicator, 

fl! = . 5X^X2 = velocity of piston in feet per minute, 

h = 1^ lbs. = effective pressure on each square inch of cylinder's 
cross sectional area, requisite to overcome friction, then. 

To find ike effective power er force of a steam engine. 

49017 ^^ effective force in horse-power. 

Example. — What is the effective force of a steam engine, the 
I diameter of the cylinder {d) being 36 inches, the stroke of the piston 
I (5) 7 feet, the effective pressure (p) 30 lbs., and making 17i revolu- 
' tions (r) per minute 1 

36-' X <3<3 — L5) X 7 X 17,5 X 2 -r- 42017 = 215.37 horse- 
. power. Ans. 

To find the nomijial power of a low-pressure or condensing engine, 

d^ X^ 
ntxfxr^ = nominal force in horse-power. 

To find the nominal power of a high-pressure, or non-condensing 

engine. 

d'^XpXa 
]9() 000 ^^ nominal force in horse-power. Or, 

d^X P X /^^ 

qIo ^^ nominal force in horse-power, the piston moving 

at the ordinary speed of 128 times the cube root of the stroke. 

To find the pressure of the steam on each square inch of the boiler'' s 

surface. 

Let P = pressure in lbs. on safety valve, 

p = pressure in lbs. on each square inch of surface, 
W = i^e/g-/i^ or resistance in lbs., as indicated by the spring 
balance. 



31U STEAM AND TILE STEAM ESOnXZ. 

w = sam of tlie weights of the lever, safetj valve, and bal- 
ancing weight, in lbs., 

5 = length of lever in inches from ita axis of naotion to a 
point vertical to the centre of the valve, 

/ = length of lever in inches from its axis of motion to W, or 
to the point on the lever at which the spring balance is attached, 
a = urea of safety valve in square inches. 
Then, 

P — tr + u; 
« : / : : W : P — ir; and =■ p. 

Example. — The length of the lever from its axis of motion to a 
point over the centre of the Siifety valve (5) is 3 inches, its length 
from tlie axis of motion to the |>onit at which the icrijL^ht or spnojK 
balance is attached (/) 24 inches, the xrrii^ht^ or pressure, ns indH 
cated bj the spring balance (W) 40 Ibg., the sum of the woiirhts of 
the lever, valve, iV:c., (w) <> lbs., and the area of the valve r».^ s«(uar6 
inches ; required the pressure of the steam per square inch . 

3 : 24 : : 40 : 320, and 

.720 + G =m 32G -1. G.5 =» 50.15 lbs. Ana. . ^ 

W : P — »r : : 5 : /. / : 5 : : P — tr : W. ^ 
P — ,n : w : : / : 5. 



To find the rohimr of steam compared with the volume of wnfrr 

Let F = elastic force of steam in pounds per square inch, 
V = volume of steam compared with volume of water. 
Then, 

24250 \ 24250 

— p— -|- 05 = V, and y^T^.^ = F, nearly ; the volume of water/ 

in every instance, being 1. j 

ExAifPLE. — The olaf'tic force of the steam is 40 lljs. to each square 
inch of the surface of its bulk ; what spjice docs it occupy, comparctl 
witli the s{):ico it would occupy if it were condensed to water ^ 

242')0 -^ 40 = O(M') -\- Of) = r)71 ; that is, when the clastic force 
of the steam is 40 pounds to eacli square inch of tbc surface of it* 
bulk, the volume of tli.» stcinn. compMnMl \\\\]\ its Vdlmnc in \v:it«T. 
is as 671 to 1. 

NoTK. — The prccctling fonr.ukis arc not sUictly correct fur all deiiiiiic.^ of sUain, l»ut 
give tlie moan of a poneral ran^rc. g 

The volume of steam compare*! with the volume of water as 1. ' - ' ' •• ' ' y ex- 
periment, when the elastic f<»rce i>» 14.7 n»>«. = 1»'».*4 ; "JO U>f'. r - 'II ; 
30 lbs. = 883 •, 35 lbs. = 767 ; 40 lbs. = 070 ; 4:> llv*. — 010 5 fh . _- 
508 ; 60 lbs. = 470 \ 65 lbs. = 437 j 70 l»»s. = 4tW \ 75 W^. = iiv. ; Mt li»>. ^- ; ♦,- •, 1*0 
lbs. =3 325 ; 100 lbs. = 295 ; 150 lbs. = 205 ; 200 lbs. — 158. . 



STEAM AND THE STEAM ENGINE. 311 

To find the temperature of the steam, its elastic force being known. 

Rule. — Multiply the 6th root of the elastic force in inches of 
mercury by 177, and subtract 100 from the product ; the remainder 
will be the temperature in degrees, Fahrenheit nearly. 

Or, if we let 

a = elastic force in inches of mercury, 

b = temperature of steam in degrees, F. 
Then, 

/^a X 177 - 100 = b, and r "[l^ J = a. 

Example. — The elastic force of the steam is 150 inches of mercury, 
(150 -r- 2.04 = 73J pounds to ea<3h square inch of the surface of its 
bulk) ; what is the temperature ? 

a7150 = 2.305 X 177 == 408 — 100 = 308^ Ans. 

Note. — The pressure of the steam in pounds per square inch X 2.04 =3 pressure in 
inches of mercury. 

The temperature due to a pressure of 14,7 lbs. per square inch = 212^ 5 20 lbs. = 228^ ; 
25 lbs. = 2410 5 30 lbs. = 252^ 5 35 lbs. = 261^ 5 40 lbs. = 269^ ; 45 lbs. = 276^ ; 
60 lbs. = 2830 5 55 lbs. = 289^ 5 60 lbs. = 296^ ; 65 lbs. = 301^ ; 70 lbs. = 306^ •, 75 
lbs. = 3110 ; 80 lbs. = 3160 5 85 lbs. = 3203 5 90 lbs. = 324o } 100 lbs. = 332^ ; 150 
lbs. = 3630 } 200 lbs. = 387o. 

To find the quantity of yjater required for steam per minute by an engine 

in motion. 

Rule. — Multiply the velocity of the piston in feet per minute, by 
the square of the cylinder's diameter in feet multiplied by 0.7854, and 
divide the product by the volume of steam compared with the vol- 
ume of water, due to the pressure exerted. 

Example. — The diameter of the cylinder is 2.^ feet, the velocity 
of the piston 245 feet per minute, and the constant pressure exerted 
by the steam 60 pounds to the square inch ; what quantity of water 
must be converted into stcaDi per minute ? 

2.52 X .7854 X 245 — 470 = 2.56 cubic feet. Ans, 

Note. — For a single-acting engine half the quantity indicated by the above rule is re- 
qmi-ed. 

To find the quantity of steam required to raise a given quantity of 
water of a given temperature to a required tempei'ature . 

The sum of the kitent and thermometrical or sensible heats of 

water in a state of vapor or steam is always the same, viz., 1178^. 

The latent heat, therefore, of vapor of water at 32° = 1178 — 32 



312 BTEAM AND THE STEAM ENCrnOf. 

= 1146; at 100° = 1178— 100= 1078°; at 212^= 1I78~?I2 
= 966°, &c. 

The latent beat of water at 32^ of sensible heat = 140^. 

If we let 

a = temperature of »team emplojed, 

b = temperature of water to be raised, 

c = tcm|>erature to which the water \» to he raised, 

d= volume of steam compared with the Yolame in water due to 
the temperature of the steam, 

w := quantity of water in cubic feet to be heated ; 

Then 
c b 

- • = fjnantit J of water (say in cubic feet) that must 

be converted into steam Iiaving a temperature o, required to raiM 

1 cubic foot of water from ^ to c. 

(c ^ b\ X fi 

^-— =1 quantity of steam in cnbic feet, at temperature a^ 

966 -|- a — c 

required to raise 1 cubic foot of water from hioe. 

r " = quantity of water in cubic feet that moit he COM 

verted into steam, and have a temperature at o, recjuired to raise? 
the given quantity of water from b to c. * 

tr X (g-— ; __ quantity of steam in cubic foet, at tempera- 

y66 + o — c ^ \ ^ i^ 

ture a, reipiired to raise the given number of cubic feet of water 

from b to c. 

ExAMTLK. — What quantity of water in steam at 212^ will rai«e 
100 cubic feet of water from 60° to 200° ? 

. — 1 00 X 140 ^^^^^ ^^^j^ ^ ^^ Making 14.81 X; 

966+212 — 200 . '^ ^| 

1694 = 24241 cubic feet of steam at 212 degrees. 

To find the quantity of xcater at a tjiven temjyeratMre required t0 
reduce a given qxtantitij of steam to a f/iven temperature. 

Let 

a = temperature of steam to be reduced, 

c = temperature to whieli the steam is to be reduced, 

/>z= temperature of the water injected. 

Then 

' , izi number of cubic inches of the water required to 

c — b ^ 

reduce 1 cubic foot of the steam from a to c nearly. 



STEAM ACTING ]EXPANSIVELY. 



313 



To find the Mean Force of Steam acting Expansively, and the Advan- 
tage Gained, 



N. 


C. 


N. 


C. 


i ^• 


c. 


N. 


C. 


N. 


C. 


1.1 


1.095 


2.1 


1.742 


3.1 


2.131 


4.2 


' 2.435 


6.3 


2.841 


1,2 


1.182 


2.2 


1.788 


3.2 


2.163 


4.4 


2.482 


6.5 


2.872 


1.3 


1.262 


2.3 


1.833 


1 3.3 


2.194 


4.6 


2.526 


7. 


2.946 


1.4 


1.336 


2.4 


1.875 


j 3.4 


2.224 


4.8 


2.569 


7.5 


3.015 


! 1.5 


1.405 


2.5 


1.916 


\ 3.5 


2.253 


5. 


2.609 


8. 


3.079 


' 1.6 


1.470 


2.6 


1.956 


1 3.6 


2.281 


5.2 


2.649 


8.5 


3.140 


; 1.7 


1.541 


2.7 


1.993 


I 3.7 


2.308 


5.4 


2.686 


9. 


3.197 


! 1.8 


1.588 


2.8 


2.030 


•' 3.8 


2.335 


i 5.6 

1 5.8 


2.723 


10. 


3.303 


1.9 


l.«42 


2.9 


2.065 


; 3.9 


2.361 


2.758 


11. 


3.398 


i 2. 


1.693 


3. 


2.099 


4. 


2.386 


i 6. 


2.792 


12. 


3.485 



S = stroke of piston in inches. 

d = distance the piston moves in inches before the steam is cut off. 

P = pressure in pounds per square inch of the steam on the piston. 

C = tabular quantity in column C, standing against the quantity in cotumn N, that 

When S -r- 6? = any quantity in column N (above table), then, if 
S -f- ^ be taken to represent the whole effect of the steam, or effect 
that ^YOuld have been produced had the steam not been cut off, the 
quantity in column C (same table) , standing against the quantity 
S -7- f/, will represent the effect, the steam being cut off as supposed ; 
that is, the effect of the steam, cut off as supposed, will be to what 
it would have been had no cut-off taken place, as the quantity in 
column C to the quantity in column N, against it. 

^p^ 

-— r- = C X 1^ -r- (S -7- 6^) = mean pressure of steam in pounds 

per square inch on the cylinder, the steam being cut off ad libitum. 

Example. — The steam enters the cylinder with a force or pressure 
of 40 pounds to the square inch, is cut off when the piston has moved 
32 inche?, and the whole stroke of the piston is 7i feet (87 inches ) ; 
required the mean pressure per square inch on the cylinder. 

87 -r- 32 = 2.7, the pressure, therefore, is to what it would have 
been, if the steam had not been cut off at all, as 1.993 to 2.7, as yinnr ? 
and the mean force or pressure on the cylinder is 

1.993 X 40 X 32 -h 87 = 29.3 ibs. per square inch. Ans. 



Steam, under a pressure of 1 atmosphere, flows into a vacuum with a velocity of about 
1400 feet a second, and into the air with a velocity of about 650 feet a second •, its velocity 
under a pressure of 20 atmospheres is about 1600 feet a second, either into the air or a 
vacuum. 

Atmospheric air at 60^, b. SO in., flows into a vacuum with a velocity 1327 feet a second. 

The velocity of a cannon ball (maximvun) is about 2000 feet a second, when near the muzzle 
of the gun, but at the distance of 500 yards from the gun it is not above 1300 feet a second. 

The greatest velocity of a rifle ball, near the muzzle of the gun, is 2012 feet a second. 

The velocity of a musket ball, full charge, 18 baUs to the pound, windage .05, near the 
muzzle of the gun, is 1600 feet a second. 

Inflamed gunpowder expands with a velocity of about 5000 feet a second. 
27 



314 Of coFTimjouB circular MonoFr, 



OF THE ECCENTRIC IN A STEAM ENGINK 

Tho throw of the eccentric is equal to twice the distance from the eenlre of 
formation to centre of revolution ; that is, it is equal to tho diameter of iho 
circle of revolution minus tho diameter of the circle of formation. If r rep- 
resent the shortci^t, and R tho longest, of ail the radii in the eccentric, meas- 
uring from the centre of tho circle of formation or axi0 of the rcTolviog shaft; 
then {li — r) X 2 = the throw of the eccentric. 

The trnrel of the ralre is equal to the sum of tho widths of the two cioam 
openings, i>lu8 a flight excess of length to the raWe more than jiist aoffieient 
to cover the openings. 

L :a length of lerer in iocbes on weigh or trarerso shaft for working the 
Talve. 

/ ac= length of loTor in inobei od weigh-ehaft for eeoenlrio ro4« 

V = travel of vaivo in inches. 

t = throw of eccentric in inches. 

W -L. L =r / . r/ -t- / =■ L ; lu -^I ^i , \^ ^ , -^f. 

To find the numUr of revolutions that each of tiro xchrtls that nr, j. arcd 
together will make Inforf thi same frith trill ronie to</tth*r ai/atn. 

Rule. — Divide the number of teeth in each wheel bj the greatest num- 
ber thai will divide both without a remainder \ the greater quotient will be 
the nunihcr of revolutions made by the smaller wheel, and the le.s!) the num- 
ber made bj the larger. If both wheels cannot be divided bj a comnioo 
divioor, the smaller wheel will make as many revolutions as there are teetb 
in the larger, and the larger am many as there are teeth in the smaller. 

Fyc.\MrLK. — Krfjuiml the numlMT of n*v(diitic)n8 inadr }»y each of 
tw(- wIkmIs that an* gt^ari'il tt^cthcr, tho l;\r;;»T having (>0 teeth, and 
the enKilh'r 21, lieforo the Hime tcvth com«' tu^ethor again. 
GO 22 revohitionB of mnalitr wheel. ) 



rcYolutions <»f lar;:«r wImi I 



OK CONTINUOUS CIRCULAK MOTION. 

When a 8<*ries of wheel?, r>r wheels, j»inion», drums and pullcyf , 
are m) arran;:^^d that one, Ix'in;; «ct in n>otion, imparts motion 
directly to another, that to a thinl, and mo on, all, at rqual d'w»""^"^ 
from their respective eentren, will (h'^eri^e equal circles of ^;. 
As are their radii, diameters, circumferenees or numlK^r ui .^ ..., 
therefore, one to another, so are their number of revolutions one to 
another, or so are their turns in the siinie space of time. 

In every machine there is some first point of impulse, or point at 
"which the motive power is applied, and the circle of gyration de- 
Bcri))ed by tliat point is the periphery of the first moved, or first 
driven, oi that machine. From some point in this circle, or fnora 
some circle descrilx^d upon the shaft which it drives, the power is 
transmitted to a remoter or next contiguous movement, thereby tho 
point so transmitting becoming the driver thereof. Thus tho whole con- 
tinuous chain consists, alternately, of driven and drivers throughout ; 



CIKCULAR MOTION. 315 

or, if we lake the motive power into account of drivers and driven 
throughout. 

Example. — A drum, on the main line of shafting, is 18 inches in 
diameter, and by means of a belt drives a pulley whose diameter is 12 
inches ; how many revolutions does the pulley make, to one revolution 
of the drum ? 

18-^12=1.5. Ans, 

What portion of a revolution does the drum make, to one revolu- 
tion of the pulley ? 

12 -7- 18 = I of one revolution. Ans, 

Example. — The above drum makes 120 revolutions a minute ; 
how many revolutions does the pulley which it drives make in the 
same time 1 

18 X 120 = 2160 -r- 12 = 180 revolutions. Ans. 
Or, 2 : 3 :: 120 : 180 revolutions. Ans. 

Example. — The diameter of the driver is 18 inches, and makes 
120 revolutions a minute ; what must be the diameter of a pulley which 
it will drive at the rate of 180 revolutions a minute? 
120 X 18 -r- 180 = 12 inches. A?is. 

"Example. — A pinion of 8 teeth drives a wheel of 49 ; how many 
revolutions does the pinion make to one revolution of the wheel? 
49 -~ 8 = 6J revolutions. Ans. 

Example. — A pinion has 8 teeth, and makes 80 revolutions a 
minute ; how many revolutions does a wheel make, in the same time, 
which has 49 teeth, and works in contact? 

80 X 8 -1- 49 = 13^3^ revolutions. Ans. 

Example. — A wheel has 49 teeth, and makes 13^^^ revolutions in 
a given time ; how many teeth must a wheel or pinion have to work 
in contact, and make 80 revolutions in the same time ? 
49 X 13 ^\ -f. 80 = 8 teeth. Ans, 

To find the number of revohitions made hy the last, to one revolution of 
the first, in a train of wheels and pifiions. 

The last wheel, or pinion, in a train, whichever it be, is neeessarily 
a driven ; — therefore. 

Rule. — Divide the product of all the teeth in the drivers by the 
product of all the teeth in the driven ; the quotient is the number or 
ratio sought. 

Example. — A wheel of 72 teeth drives a pinion of 14, upon 
whose shaft is a wheel of 56 teeth that drives a pinion of 10, upon 



316 CIBCnLAB MOTION. 

whose shaft is a wheel of 35 teeth that drives a pinion of 6 ; how 
many revolutions does the last pinion make, to one revolution of the 
first wheel ? 

72 X 56 X 35 

1^ Ky in v^ r = 168 revolutions. Ans, 

14 X 10 X o 

Example. — A wlicel of 72 teeth drives another of 29, upon whose 
shaft is a pulley of 21 inches diameter that gives motion to one of 10; 
what niinibcr of revolutions are made by the last pulley, to one revo- 
lution of tlie first wheel ? 

72 X 24 

^— — — ^ 5.96 revolutions. Ajis, 

The rule is Well established, that, in training for the purpose of 
accumulating velocity, or for the purpose of diminishing an over ac- 
cumulated, a certain mean or proportional velocity, between the sev- 
eral movers, should exist; and, further, that without some important 
reason for a Ingher ratio, the numlKir of icolh on the wheel should not 
exceed 6 to ] on the pinion with which it works. 

The mean, or projK»rtional velocity alluded to, is found by the fol- 
lowing rule, and is applied as shown in Kxamplks. 

RiLK. — Multiply the given and required velocities toother, and 
extract the square root of the product, which is the roeao sought. • 

Example. — From a wheel of 72 teeth, making 20 revolutions a 
minute, motion is to he conveyed, hy help of two intermediate wheels, 
to a pinion having 15 teeth, which is required to make 120 revolu- 
tions a minute, or to 1 of the first wheel ; what number of teeth 
should he on encli <»!' the intermediate wheels? 

\/l20 X 20 = IS mean velocity. 
72 X 20 -1- 49 = 29. 1 teeth on 1st driven. > , 
120 X 15 ~ 49 = 30.7 teeth on 2d driver. J ^^^' 

^''''''*- [5 X 29 ^^ ' ''''' iTxl^* ^ '' *^' ^ 20= 119-1- revolutions. 

Example. — A wheel of 72 teeth, making 20 turns a minute, is to 
drive another, on whose shaft is a wheel of 30 teeth, that is to drive a 
pinion at the rate of 120 turns a minute ; how many teeth must be OD 
the intermediate wheel, and how many on the pinion? 

a/ 1^0 X 20 = 49, and 

72 X 20 -7- 49 = 29 teeth on intermediate wheel. > ^ 
30 X 40 -^ 120 = 15 teeth on pinion. ^Ans, 

Example. — A wheel having 72 teeth, and making 20 revolutions a 
minute, is to drive another wheel, on whose shaft is a pulley of 28 
inches diameter, from wiiich pulley motion is to he conveyed to an- 
ollier pulley, required to make 120 revolutions a minute ; what nam- 



CmCTTLAR MOTION. 317 

oer of teeth should the intermediate wheel have, and what must be 
the diameter of the last pulley 1 

a/i20X20 = 49, and 

72 X 20 -7- 49 = 29 teeth on intermediate wheel. > . 
28X49 -T- 120=11^3 inches diameter of pulley. J^^^- 

The distance , from centre to centre, of two wheels to work in contact j 
given, and the ratio of velocity between them, to find their requisite 
diameters. 

Rule. — Divide the given distance by the given ratio, plus 1, and 
the quotient will be the radius of the smaller wheel ; subtract the 
radius of the smaller wheel from the given distance, and the differ- 
ence will be the radius of the larger, which, multiplied by 2, respect- 
ively, gives the required diameters of each. 

Example. — The distance from centre to centre of two shafts is 28 
inches, and one shaft is required to make three revolutions while the 
other makes one ; what must be the diameters of the wheels which 
turn the shafts, (measured from their pitch lines,) to produce the 
required effect ? 

28 -1- 4 = 7, and 28—7 = 21 ; hence, 

7 X 2 = 14 in., diameter of smaller. > . 
21 X 2 = 42 in., diameter of larger. J ^^^' 

Example. — The distance from centre to centre of two shafts is 40 
inches; one makes 44 turns a minute, and the other is to make 110; 
what must be the diameters of the wheels at their pitch lines ? 
110 -h 44 = 2.5 ratio or mean velocity ; then, 
40 -r- 2.5+1 = 11.43 X 2 = 22.86 in. > . 
40 — 11.43 = 28.57 X 2 = 57.14 in. ^^^** 

To find the velocity of a belt. 
The velocity of a belt is equal to the surface velocity of any drum 
or pulley over which it runs, or which it turns, in equal times and 
terms of measurements. 

Example. — A drum whose diameter is 6 feet makes 120 revolu- 
tions a minute ; what is the surface velocity of the drum per minute, 
or what is the velocity of the belt per minute 1 

6 X 3.1416 X 120 = 2262 feet. Ans. 
Velocity of belt -r- revolutions of drum = circumference of drum. 
Velocity of belt -r- circumference of drum = revolutions of drum 

To find the draft on a machine. 
General Rule. — Multiply, continuously, all the driven wheels, 
by way of their teeth, and the diameter of the front roller, together, 

27* 



318 TEETH OF WHEELS., &C. 

and, in like manner, all the drivers, by way of their teeth, and the 
diameter of the brick roller, together, and divide the former j)ro(l\ict 
by the product of the latter ; the quotient is the draft. 

Example. — Tlie driven wheels of a drawinij frame head have, 
one 72 teeth, the other 10, and the diameter of the front roller is l^\y 
inches ; the drivers have, one 25 teeth, the other 30, and the diameter 
of the back roller is ^ of an inch ; what is the draft ? 

72 X 40 X I.IH- (30 X 25 X .9) = 4.694-. Ans 

ExA.MPLK. — The pinion on the front roller of a spinnin«r iranic nas 
40 teeth, and the diameter of the roller is 1^ inch ; the wheel on the 
back roller has 50 teeth, and the diameter of the rolh»r is 5 <»f an inch; 
the stud gears have, driver, 21 teeth, driven, 84 ; wliat is the draft ^ 
84 X 50 X 1.^-7- (40 X 21 X .75)= 10. Ans. 

To find the rei'oluiions of the throstle spindle. 

RuLK. — Multiply the diameter of the cylinder by the number ot 
rev(jluiions it makes in a given time, and divide the prmlucl by the 
diameter of the whir ; the quotient will \ni ilic nunibLT of n-vuhiiioiia 
of the spindle made in the same time. 

Example. — The diameter of the cylimlcr is h mciics, and n 
480 revolutions a minute ; the diameter of the whir is I of an 
how Tuany revolutions are made by the spindle per minute ? 
480 X 8 -f- .875 = 4388.G revolutions. Ans, 

To find the number of twists per inch given to the yarn by the throstle 
lluLK. — Divide the numl)er of revolutions of the spindle, in any 
given time, by the number of revolutions of the delivery (front) 
roller, multiplied by its circumference, in inches, niade in the sams 
time. 

Example. — The diameter of the front roller is | of an inch, and 
makes 110 revoluti(»ns a minule ; the spindle nnolves 43H8.6 times in 
a minute ; what number of twists per inch has the yarn ? 

.75 X 3.1110 = 2.:J5()2 inches circ. of roller; then, 
4388.0-^(110 X 2.3502)= 17 twists, nearly. Ans, 

TKKTH OK WHEKLS, &.C. 

explanations. 

Pitch Lim. — A circle defining the Ixise of the workin<; or impinging teclion of Uw 
teeth. 
Piicfi of a Wlietl —The distance from centre lo centre of two adjacent teeth, i 

ured up.in their pitch line. 

L>nii(h of a Tooth. — The distance from it^ base to its extremity. 

nniuith'ofa Tooth. — The Icnclh of tho f.icc <»f the wheel. 

Thickness of a Tooth. — The chord of the arc deacritjcd upon it by tho pilch line ; i 
greale^it cross section to tlie breadth. 



Pitch X 2.5 = breadth. 
Thickness X 1.5343 = length. 
Thickness X 2. 1277 = pitch. 



TEETH OF WHEELS, ETC. 319 

Circumference of a Wheel. — The pitch line, and its diameter is measured therefrom. 
See diagram — Geometry. 

Pitch X .^Y = tliickness. 

Length X .65154 = thickness. 

Thickness X 5.31925= breadth. 
Pitch X number of teeth X .31S = diameter. 
Diameter -^ pitch X -318 = number of teeth. 
Diameter -7- number of teeth X -318 = pitch. 

As the number of teeth on the wheel, -j- 2.25, are to the diameter 
of the wheel, so are the number of teeth or leaves on the pinion, 
4" 1.5, to the diameter of the pinion. 

Example. — A wheel, 16 inches in diameter, and having 81 teeth, 
is to pitch with a pinion having 25 leaves ; wiiat must be the diameter 
of the pinion? 

81 + 2.25 : 16 :: 25 + 1.5 : 5.093 inches. Ajis. 

As the number of teeth on the wheel, -j- 2.25, are to the diameter 
of the wheel, so are half the number of teeth on the wheel, -\- half 
the number of leaves on the pinion, to the distance their centres should 
have. 

Example. — Awheel is 16 inches in diameter, and has 81 teeth; 
the pinion with which it is to work has 25 leaves ; what should be 
the distance from the centre of the wheel to the centre of the pinion ? 
81 + 25 = 106 -^ 2 = 53 ; then, 
81 + 2.25 : 16 :: 53 : 10.1862 inches. Ans. 
As half the number of teeth on the wheel and pinion are to the dis- 
tance from centre to centre of the wheel and pinion, so are the num- 
ber of teeth on the wheel, -{- 2.25, to its diameter; or, so are the 
number of teeth on the pinion, -(- 1.5, to its diameter. 
53 : 10.1862 :: 83.25 : 16 inches. 
53 : 10.1862 :: 26.5 : 5.093 inches. 

As the number of teeth on the wheel and pinion, — 3.75, are to the 
distance between the centres of the wheel rnd pinion, so is the cir- 
cumference of the wheel to the diameter of the pinion, very nearly, 
and proving the almost strict accuracy of the foregoing. 

106 — 3.75 : 10.1862 :: 16 X 3.1116 : 5.0075 inches. 

To find the horse power, at a given veloeiii/, of a cast iron tooth con- 
structed on the foregoing principles. 
Rule. — Multiply the breadth of the tooth, in inches, by the 
square of its thickness, in inches, and divide the product by twice the 
length, in inches; the quotient, multiplied by the velocity in feet per 
second, gives the reliable horse power at the velocity specified. 

Example. — The teeth on a wheel have each a breadth of 10.64 
inches, a thickness of 2 inches, and a length of 3.07 inches, 



L 



320 HYDROSTATICS. 

required their reliable strength, in horse power, at a velocity of ^ 
feet per second. 

10.64 X 2^ X 6 -r- 3.07 X 2 = 41.59 horse power. Ans. 

JOURNALS OF SHAFTS. 

To find the requisite diameter of a cast iron journal to resist torsion and 
stress in overcoming a given resistance at a given velocity, 
Mr. Buchanan pives deductions, from which are derived the following 
rule, for asccrlaininj^ the proper diameter of the journal of a water- 
wheel, or first mover, in any machine. 

Rule. — Multiply the resistance, in horse power, by 400, and 
divide the product by the number of revolutions of the wheel per 
minute ; the cul)e root of the quotient is the requisite diameter of the 
journal, in inches. For shaftij inside of the mill, to drive smaller 
machinery, use 200, instead of 400, for the muliiplier, and, if the 
shafts are to drive still smaller machinery, use iOO as the multiplier. 

Mr. Grier pives the mean of all these, or 240, aa Uie multiplier, to 
resist torsion alone, and directs to take, for second movers, the diam- 
eter thus found, muliiplied by 0.8, and for third movers the same 
diameter multiplied by .7t*3. 

If the journal is wrought iron, multiply the diameter, found by the 
preceding rule, by .903 ; if of oak, by 2.238. 

KxAMTLK. — The diameter of a water-wheel is 16 feet; the resist- 
ance it has to overcome (at its pitch with \\\c jack) is 40 horse |)ower, 
and the surface velocity of the wheel is feet per second ; what should 
he the diameter of its journals ? 

CO X 6 -r- 10 X 3.1410 = 7 revolutions of wheel per minute ; and 
40 X 400 -f- 7 = ^2280 — 13.2 inches. Ans. 



HYDROSTATICS. 

All fluids, at rest, press equally in every direction. The pressure 
exerted by them, therefore, can never be .so little as their weipht, and 
may, under circumstances, be to almost any conceivable extent 
g^reater. The ddwuward pressure exerted by a fluid is its weight, 
and its weight is as the quantity; hut the lateral pressure exerted is 
in a measure independent of quantity, being dependent upon depth, or 
vertical height. 

Any given area, in any given section of a containing vessel, is 
pressed equal to the weight of a column of the fluid whose base is 
equal to the area pressed, and whose height is equal to the distance 
of tlie centre of gravity of that area, below the surface of the fluid ; 



HYDROSTATICS. 321 

this IS the case whether the sustaining surface be horizontal, or ver- 
tical, or oblique. 

The bottom of a containing vessel, therefore, whatever be its shape, 
sustains a pressure equal to the weight of the superincumbent fluid, 
or equal to the weight of a column of the fluid whose base is equal to 
the area of the bottom, and height equal to the distance from the bot- 
tom to the surface — equal to the area of the bottom, multiplied by 
the depth of the liquid, multiplied by its weight, in like terms of 
measurement. 

And each side of the containing vessel, whatever number of sides 
there be, sustains a pressure equal to the area of that side multiplied 
by half the depth of the liquid, multiplied by its weight, in the same 
terms of measurement. 

Thus, a rectangular vessel, whose sides and bottom are equal, and 
each two feet square, has a capacity of 8 cubic feet ; it will hold, 
consequently, 8 cubic feet of fresh water, one cubic foot of which 
weighs 62i lbs. It will hold, therefore, 62J X 8 = 500 lbs. of water. 
Now, if we suppose this vessel filled with water, we have, according 
to the foregoing, a pressure on the bottom of 2 X 2 X 2 X 62.5 = 
500 lbs. ; a pressure exactly equal to the weight of all the fluid. And 
we have, upon each of the four sides, a pressure of 2 X 2 X 1 X 
62.5= 250 lbs. ; a lateral pressure, therefore, equal to 250 X 4 = 
1000 lbs., — equal to twice the pressure on the bottom, and showing 
the entire pressure exerted to be 300 per cent, greater than the weight 
of the water employed. 

Again : if we suppose the above vessel contracted, laterally, to the 
extent that its sides are but 3 inches, or J of a foot apart, through- 
out, and that its length is so extended that it still holds the 8 cubic 
feet of water, then v^^e have, upon the bottom, whose area is only 9 
square inches, a pressure of .25 X .25 X 128 X 62.5 = 500 lbs. 
as before ; and upon each side we have a pressure of .25 X 128 X 
A2 8 X 62.5 = 128000 lbs. ; making in all a pressure of 128000 X 
4 -}- 500, — the enormous pressure of 512500 lbs. ; and that too ex- 
erted by 8 cubic feet or 500 lbs. of water. It is easy to see that the 
same principles hold good under any extent of lateral area. 

Example. — A sluice or flood-gate is 3 feet by 2^, and its centre is 
12 feet below the surface of the water ; what pressure does the water 
exert upon it 1 

3 X 2.5 X 12 X 62.5 = 5625 lbs. Ans. 

Example. — A dam, that presents a perpendicular resistance to a 
stream, is 40 feet long and 15 feet high; the water is level with its 
top ; what pressure does the dam sustain, supposing the water at rest, 
and what is the mean pressure against it per square foot ] 
40 X 15 X y^- X 62.5 = 281250 lbs., pressure against the dam ; and 
281250 -T- 40 X 15 = 468 J lbs., mean pressure ggr sq. foot. Ans. 



322 HYDRAULICS. 

Example. — The same stream, the same length of dam, and iho 
same vertical height as the precedin^r, and the dam slopintr into the 
stream against the current, oO rect from its hase ; required the pres- 
sure against the dam, and the average pressure {HJr square foot. 
40 X 15 X 7.5 X (i2.5 == 281250 Ihs., press, as before. n 

Vl5^ -f- 3 0^ = 33.541 feet, slant height of dam ; and ( Arts. 

281250 4- 40 X 33.541 = 209.63 lbs. av'g pres. per sq. foot. ) 



HYDRAULICS. 

The establishrd law for the velocity of all bodies falling from rest 
is given iindcrCtRAViTATioN, viz., that y h«^Ji:ht X 04.33, or ^height ' 
X 8.02 = vrlocity per spcond, or veh»city in our second of time, the ' 
velocity and height both being in the same denomination of measure. ' 
And from what has been said concerning pre.ssur(% under Hydro- 
statics, it is evident that the s;une law will e:ius<> water, or other 
fluid, to flow through an opening in the side of a reservoir, or dam, 
witfi the same veloeiiy that a body wtiuld attain falling perpendicu- 
larly through a .«*pare ecjual to that lu'tween the surface of the water 
and the centre of ihe o|)enintr alluded to ; ami that, con.sequenlly, the- 
oretirally, the quantity thus di.Hcharged, in any given time, will l)0 
equal to the product of the velocity and area of the ojwnlng, mulli- 
jdied by tiiat time. 

The theoretical law, however, last adduced, under ordinary circum 
stances, does not apply. And the quantity di.Hrharged, owing io the 
contraction of the thud vein, caused by tile friction of the particles 
against the sides of the openiriLf, falls short of that theoretically due. 
The only instance known in which the full force of the law may Ikj 
obtained, is where the discharge is made to issue through a straight 
tube whosi> form is the frustum of a cone, \\» length IxMng half the 
dianuMer of the aperture, and the diameter of the receiving end to 
that of the discharging end as 5 to 8 ; when a fluid is allowed to pass 
through such an opeuiuir, no contraction of the vein takes place. 

From various carefully conducted experiments by M. Morin, Eytcl- 
wein, JJossut, and others, the following practic^il rules for ascertain- 
ing the quantity discharged through dilierent openings, and under dif- 
ferent heads, are derived : — 

1. When the issue is through a circular oi)ening, its upper vertical 
point as high as the surface of the fluid, estimate the height or head 
from the centre of the opening to the surface of the fluid, and use 5.4, 
instead of 8.02, as the coelficient of quaiitity. 

2. When the opening is circular, and under a head equal to its 
diameter,' estimate the head as in the preceding, and use 8 as tho 
coefficient. 



WATER-WHEELS. . 323 

3. When the issue is through a rectangular orifice, two or more 
l^et beneath the surface, estimate the head from the centre of the ori- 
fice to the surface of the water,, and use 5.1 as the coefficient. 

4. When the discharge is from a rectangular opening, extending as 
high as the surface of the fluid, estimate the head from the bottom of 
the opening to the surface of the water, and use 3.4 as the coefficient. 
This rule applies to water flowing over a dam, or from a notch or slit 
cut in its side, &c. 

It may be proper to add, that if the orifice is small and under con- 
siderable head, the quantity discharged, relatively, will be slightly 
less than would be discharged if the opening w^ere nearer the surface. 

From the foregoing we obtain the following 

GENERAL RULE. 

Multiply the square root of the height, or head, (as estimated in 
the foregoing,) in feet, by the coefficient of quantity given as per- 
taining thereto, and the quotient will be the eflective velocity in feet 
per second of the discharf^e ; which, multiplied by the area of the 
opening in feet, gives the quantity in cubic feet discharged in a single 
second, or in each second of time. 

Example. — A rectangular opening in the side of a dam is 6 feet 
long and 8 inches deep ; and the distance from the centre of the open- 
ing to the surface of the water is 4 feet ; required the quantity of 
water discharged in each second of time. 

V-i = 2X5. 1x0X5= -10.8 cubic feet. Ans. 

Example. — A dam is 60 feet long, and the water flows over its 
entire length 6 inches, or h foot deep ; what quantity flows over per 
second? 

a/. 50* = .7071 X 3.1 X 00 X .5 = "^2^ cubic feet. Ans, 



WATER-WHEELS. 

The many uncertainties and doubts which existed until lately con- 
cerning the best mode of constructing a water-wheel, with the view 
to obtain a maximum of elfect, the velocity at which thew^heel should 
move, and other requisites pertaining to it generally, have, in a great 
measure, through investiirations and experiments by the Franklin In- 
stitute in this country, added to those by M. Morin, in France, and 
other parties interested, been removed ; and the whole seems now to 
have nearly subsided into the following general and demonstrative 
conclusions : — - 

* The decimal .50 is ihe same value as .5, = i, bul it will be recollected that to obuia 
the root 0/ a decimal full periods must be used. 



324 • WATER-WHEELS. 

1. That, to obtain a maximum of efTecl by a horizontal water- 
wheel, the water must be laid upon the wheel on the stream side, and 
at a point or line on the wheel about 52J decrees distant from its 
mit; or, the effective fall — distance from the centre of the dis« 

in^ orifice to the bottom of the wheel pit, or water therein, when mo 
wheel is at rest — being 1, the diameter of the wheel should Iw 
1.108. 

2. That the periphery of the wheel ought to move at a velocity in 
feet per second equal to about twice the square root of the number 
of feet eflfective fall ;• that the number of buckets should equal 2.1 
times the wheel's diameter in feet ; and that due means be adopted for 
the escape of the air from the buckets, either by causing the stream 
to flow some inches narrower than the wheel, or otherwise. 

3. That a head of water is requisite sufficient to cause the velocity 
of its flow to be as 3 to 2 of the velocity of the wheel ; about JL 
of the whole fall, or entire height, being equal to the required head. 

4. That a wheel of good workmanship, constructed and geared 
according to these restrictions, will return, as a maximum, about 80 
per cent, of the power employed. 

5. That, because of water producing a less effective power by im- 
pulse than gravity, turbines, or wheels through which the motion is 
obtained by reaction, are greatly preferable to undershot or low-breast 
wheels ; that they arc, seemingly, as well adapted to great as to small 
falls, return in«r, in either instance, under favorable circumstances, an 
useful effect of from 70 to 78 per cent, of the power expended ; that 
their velocities may vary considerably from that affording the maximum 
effect; (g of their light velocity,) without materially diminishing their 
effect; that they are nearly as effective when drowned to the depth of 
several feet as when working free, thereby making use of a ^'i- •• r 
fall than can l)e obtained, at the same locality, for any other ui. . i . 
and that they receive variable quantities of water without altering 
the ratio of the power to the effect. 

Th«^sc considerations, taken in connection with the less important, 
that they are durable, not more liable than others to require repairs, 
occupy less room in their position, and cost decidedly less than other 
motors of e«iual efficacy, are f:ist hrinnrinir this clnss of wheels into 
fiivor and use. 

To Jind the puicrr nj a .<ir'i!:n i > ua/i iiii ■'''•-■-!. 

Example. — The entire heichl— heat! ami fall ic nutn- 

l;ty of water flowinjj '\s that which may Iw ilravvn t - . .-; 16 feel 

long^ and 3 inches deep; n^v'*''^^ ^^® grealosi exeriive lorco o( lite slroain, in boree- 
po^er, applied as alxive suppiised. 

IC X 12 X ^-r ^^ ' = * ^'^•'l area of discharge. 
I'J X ' — I = - •^"■^ ^eet retniifile head. 
^2.32 = 1.523 X ^ I = "'6 feel, effective velocity per second. 

^'Tho practice la l^econiing very general to gear all wbeelii great or fmall, to a Telocity 
of about 6 foet per second. 



WATER-WHEELS. 325 

7.76 X 4 = 31 cubic feet discharged per second. 
31 X 60 = 1360 cubic feet discharged per minute. 
1860 X 62.5 = 116250 lbs. discharged per minute. 
116250 X ly — 2.32 = 193'J050 lbs. momentum. 
1939050 -r 33000 = 56i horses' power. Atvs. 

To find the requisite dimensions of a wheel, based upon the preceding principUu, 
adapted to the foregoing 'stream, 

19 — 2.3ii= 16.63 feet effective fall. 

16.68 X 1.103 = 18.48 feet diameter of wheel. 

360° : 13.48 : : 52^.75 : 2.7 feet of wheel above discharging orifice. 

18.48 — 2.7 = 15.78 feet of wheel below discharging orifice. 

16.63 -- 15.78 = 0.9 foot clearance of wheel. 

^y 16.63 = 4.03 + X 2 = 8. 16 feet velocity of wheel per second. 

18.48 X 2.1 =39 buckets. 

18.43 X 3.1416 = 53.06 feet circumference of wheel. 

8.16+ X 60 == 490.17 feet velocity of wheel per minute. 

490 -7- 53 =: 8.44 revolutions of wheel per minute. 

1360 li- 490 = 3.8 feet sectional area of buckets. The buckets, to properly retain tho 
water and avoid waste, should be but half full ; therefore — 

3.8 X 2 = 7.6 feet practical sectional area ; and to allow sufficient room for the escape 
of the air, the wheel should be, say 9 feet broad. 

7.6 -^ 9 = .35 — , isay 1 foot depth of shrouding. 

18G0 -^ 39 X 3.44 = 5.63 cubic feet of water received by each bucket. 

53 _i_ 39 =: 1.49 foot breadth of bucket. 

5.63*-^ 1.49 = 3.3 feet sectional dimensions, as before. 

3.8 X 2 X 1-'^^ = 11.36 feet practical capacity of buckets, more, allowance as above. 

From a well constructed wheel the water begins to empty at about 5 feet from the bot- 
tom ; there fore — 

13.43 _ 2.32 -f-. 5 -[-5 = 10.66, ratio of diameter to loaded arch. 

18.43 : 4-9 : : 10.66 : 11.2 loaded buckets. 

11.2 X 5.63 X 62.5 = 3971 lbs. on loaded arch. 

Or, a very good and safe rule for determining the weight, in pounds, constantly on the 
wheel, is this : — Multiply -^- of the buckets on the wheel by the number of cubic feet of 
water received by each, and that product by 40. Ex. 

39 -^ 9 r= 43^ X i X 5-63 X -10 = 39:3s lbs. load. 
3971 X ■^•^^ = 1943790 ll)s. ; or, } momentum, or 

39- X 3.44 -hX 5-6^- X 6:^.5 X 16.&3 = 1939050 lbs. \ exertive force. 
19:39000 -^ 3:3000 = 53.75 X -^^ = 44 h. p. effective. Ans. 

From the foregoing a very good practical rule is derived for determining the requisite 
head for any given velocity of wheel ; thus — 

height -{- 3.16 : 2.:32 :: heiyhi -|- required velocity : required head. 

Example. — The entire height is 16 feet, and the velocity of the wheel is to be 6 feet 
per second ; required the necessary head. 

16 -f 6.16 : 2.32 : : 16 -f- 6 : 2.11 feet. Ans. 

It has been denivjusirated that in practice nearly two feet head is required to generate a 
velocity of 5 feet per second, and this rule gives 

10-f 3.16 : 2.:32 : : 10 -f 5 : 1.916; or, 25 -f 3.16 : 2.32 : : 25 -|- 5 : 2.099, showing a 
ditfereiice of otdy .13, or a mean error of about one inch, between the two extremes — 10 
feet head and fall, and 25 feet head and fall. 

If it were intended to construct a wheel for the foregoing stream, to run at a less veloc- 
ity, say at 6 feet per second, (the other restrictions to be maintained,) then we should 
have 

6 X 60 = 360 feet velocity of wheel per minute. 

360 -^ 53.33 = 6. 12 revolutions of wheel per minute. 

1860 -^ 360 = 5 feet sectional area of buckets. The depth of the buckets should sel- 
dom vary much from 12 inches; the required breidlh of the wheel, therefore, in thi.i 
instance, would be about 11 feet. But it is customary, perhaps unadvisedly, when tho 
Telocity is intended at 6 feet or less, to place the buckets a little nearer together than com* 
28 



326 DYNAMICS. 

ports with the preceding; in which case, of course, a less breadth of wheel 
would be required. 

If we multiply the number of cubic feet flowing upon the wheel per rainote, 
by the effective fall, and divide the jjroduct by 700, we obtain the effectlTe 
horse-power; thus — 

1800 X lC.GS-!-700 = i4.3 horses' power. 

The power should be taken from the wlieel on the Ride to which the water is 
applied, and at a point liorizontal to the centre of the wlieel; and Ix'lts. in- 
stead of gears, for all the prime movers, should be used aa far as practicable. 



DYNAMICS. 

UNIFORM MOTION. 
P -zz. mechanical |K)wer in pounds, or i)ower in efTccts. 
F =z force in jwunds, or wci<;ht or rc^isUince to bo overcome. 
V = velocity of the Ibrco F in feet pt»r second. 
t = time in seconds during which F is in motion. 
« = space in foet thmugh whicii F moves in the time L 
m = momentum of the li)rce /•' in the time /. 
M = mass, or moving matter in eilects. 
W = work in foot-pouuda of jKjwer. 

P = FV=Fs^t. F=P-^V=Pt^8. V=P-^F = s-^L 
t = 8^V=Fs^P. 
8=.Vl = Ft--rl\ m = Ft = MV, 3/=/V^r = >K-f-r. 
W= Fs = Ftv = MV^= PL 

I NIFORMLY ACCELERATED MOTION. 

F=V-^gt z= 2s -^gez=i V^ -^ 2gs. V= 23-^1 = Fgt =\f{2gFs). 
s=zi(v = yFt' = V'^2gF. t=i2s-^V=V^gF=^(8-^{gF). 

MOTION OVER A FIXED PULLEY. 

Let A and a represent the opixwiug forces or weights. 
g=32.lG(j feet per second. (See p. 85.) 

HYDROSTATIC PRESS. 

A = transverse area of cylinder piston. 
X^ = diameter of cylinder piston. 

a = transverse area of foiving-pump piston. 

f/= diameter of forcinu-punip piston. 

h =r diameter of safety-valve. 

/= pressure in pounds on safety-valve that prevents it from 
rising. 

a:A::F:P. cPiu^iif-.p. D'li^y.P:/. 



SECTION VI. 

COVERINGS OF SOLIDS OR PROBLEMS IN 
^ PATTERN CUTTING. 



Under this head, I propose cliiefiy to contemplate patterns that 
are applicable to the wants and purposes of Tin-Plate and 
Sheet-Iron Workers ; and, in treating of the construction of these, 
my main purpose will be to offer clear and unmistakable step-by- 
step dire(;tions for constructing them practically^ and, as far as ad- 
missible by mechanical means ; to the end that the student unac- 
quainted with the principles involved in his tasks, and reluctant to 
enter into mathematical calculations, can nevertheless accompHsli 
his purposes, and with accuracy and despatch : and I shall accom- 
pany the proceedings with diagrams for illustration and reference. 
But since I shall be obliged to view the patterns theoretically and 
analytically in all their parts, in order to devise the best rules for 
constructing them practically, I shall deem it not unadvisable, with 
reference to many of them at least, to state the laws and the math- 
ematical data upon which the directions for their construction will 
be predicated. Moreover, many of the patterns will be found of 
a high geometrical type, governed by inflexible laws, and capable 
of mathematical investigation and measurement in all respects ; 
and to present these only in their naked aspects of mere mechani- 
cal contrivances, to be fixed in the memory, or copied at will, 
would seem out of place in a work of this kind. 

Problem 1 of the following series will embrace in its solution 
all the principles involved in the construction of the whole class of 
patterns to which it will relate ; and, more or less interwoven with 
its solution mechanically, I shall, once for all, with reference to 
the class, enunciate the laws and define their bearings, to the full 
end of constructing them theoretically and mathematically ; and I 
shall do this in as brief a manner and as free from technicalities as 
the circumstances will admit of. 

But, before proceeding to the solution of problems, it may be 
proper, perhaps, to explain the meaning of some terms that I shall 



328 PATTERN CUTTING. 

be liable to make use of; and it may as well be done hero, per- 
haps, as elsewhere. 

A vessel in the form of a frustum of a cone or truncated oooe, 
is dijlunng vesst^l haviu^^ cireles for its bases. 

The lateral portion of a conical vessel or cylindrical velBel is 
the sidfi or body. 

The bases of a vessel are the ends. 

The fixed base of a vessel is called the bottom ; and the moTi^ 
ble base, the cover. 

The slant (le[)th of a vessel (and noue but flaring Teasels have 
slant depths) is the depth of the side. 

The diameters of a vessel are the diameters of the bases or 
ends. 

The perpendicular depth of a ve!«el is its depth. 

The circumference ol a circle is e<iual to the diameter multi- 
plied by 3.141G ; or it is C(|ual to the diameter multiplied by *«.*>.>, 
and divided by 113 ; or it is nearly etjual to the diameter multi- 
plied by 22, and divided by 7. 

The diameter of a circle* therefore, is c<]ual to the circumfer- 
ence divided by 3.1 4 IG; or it is erpial to the circumference mul- 
tiplied by 1 13, and divided by 3.'>r> ; or it is nearlv equal to the 
circumference niulliplie<l by 7, and divided by 22. 'the (ireek let- 
ter n-, if met with in connection with the problems, will invariably 
mean 3.14 IG, or the ratio of the circumkrence to the diameter, 
the latter hc\\v^ 1. 

O = Solidity ; A = Area ; U = Curved or convex surface of a 
Solid. 

In geometry, written lines are limited by the letters or charac- 
ters tliat are placed at their extremities ; and, in the text, they are 
announced by the same letters or characters written with a space 
btitween them. Thus a 6, X: m, c r, iVc, in the text refer to the liius 
limited by a b, I m, c z, tSr., in the diajrrams ; but the values of th« *e 
lines, that is their lengths, when they ^rv. intnj<liiced into e.j .,i- 
tions by the letters that limit them, .ire otherwise expressed: tims 
a 6 or (a b) in the text or an ecpiation means the length of the line 
a 6, or that is limited by a and b; ab or (a by means the s^piare of 
the length or line ab ; 2 a b or 2(a />), twice the length of the line 
a by Ssw 

In algebraic notations, factors and numeral coK'flicients and 
factors are usually written without the sign of multiplication or 
a space between them; thus af>Cj 2(i, J2a^>, (nb — iI)(cd-\-m), 
are to be read, a X 6 X ^, 2 X ^^ 12 X a X ^» (« X b--d) 
(c X <^-|-''0» that is, the difference of </ and the product of a into b 
is t(^ be multiplied by the sum of the product of c into d and the 
quantity m. 



PATTERN CUTTING. 



329 



pROB. 1. — To consfruct n Pattern for the Lateral Portion of a ves- 
sel in the form of a Frustum of a Cone of (jiven Diameters and 
Depth. 

The chief principle involved in the construction of this descrip- 
tion of patterns is easily explained : it is that of a riglit cone 
placed upon its sid(.', and rotatinir on a plane. If a cone so placed 
and startinin^ from rest make one revolution, or, in other words, roll 
once over, its whole lateral surface, correctly delineated, may be 
supposed to be described upon the plane ; if it make a half-revolu- 
tion or roll half over, half its lateral surface in like manner delin- 
eated may be supposed to be described ; and so on for any partial 
or fractional rotation whatever : thus the slant height of the rotat- 
ing cone will be the radius of the arc that will be described by the 
rotating base, and the arc so described will be that of the lateral 
surface, lateral portion, side, body, or covering. The rotatinf]^ 
com?, then, that will describe the greater arc of the lateral surface 
of a frustum, must be a cone including the frustum ; and that that 
will describe the lesser must be the same cone, less the frustum. 

Rule. — Place the square suitably on the plate from which the 
pattern is to be taken, as 
a m Rj diagram, and draw 
to any sufficient length a 
line 771 R; and from ?;?, on 
the other arm of the square, 
set olF a known measure 
(the whole or any desired 
aliquot part, as J, ^, \) of 
the greater given base, m a, 
and draw that measure ; 
then drop the square per- ■ 
pendicularly down on the 
line m R, from 7?j, equal to 
the given depth of the ves- 
sel, 7)1 h, and set off in like 
manner the same known 
measure of the lesser given 
base, h c, and draw that 
measure ; then draw a lii^, 
a c R, through the points 
a and c, to the line m /t, 
and cutting that line in R. Next, with the distance R a m mtj ui- 
viders or on the beam compasses, and R the centre, describe an 
arc, as a b, to any sufficient length ; and with the distance R c, and 
28* 




330 PATTERN CUTTING. 

R the centre, describe a parallel arc, as c cf, to any sufficient 
lenfrth. 

We have now defined the curves for the level surfaces (top and 
bottom) of the given vessel, and the depth of its sides ; or, in other 
words, we have defined tlicr rntio of the given diameters and the 
slant depth of the vessel ; and have thus far a pattern, in some 
sort, for a vessel of this general form, having the same slant depth, 
the same ratio of diameters, and the diameters varying from al- 
most nothing to almost twice the radii to which the respective arcs 
have been drawn ; or, in another point of view, having the same 
slant depth, tlie same ratio of diameters, and a depth varying from 
almost nothing to almost the depth of the side. A right section 
taken out indiscriminately, a c r rj, for example, would l>e a pat- 
tern for this kind of vessel at a fixed slant depth and a fixed ratio 
of diameters, and such patterns are sometimes used ; but clearly 
it Avonhl be no pattern for a vessel of a givi'n perpendicular depth 
and given diameters, since it would Ixi no known measure of iho 
lateral surface recpiired. The slant depth and the ratio of the 
diameters remaining constant, the j>erpendicular depth varies, as 
we have seen, with the diameters ; and the given diameters, it ap- 
pears, have as yet in no degree Ix^en fixed or defined, only their 
ratio has been defined. Hut .since the perjK^ndicular depth varies 
with the diametei-s, and the ratio of the given diameters has been 
defined, it follows, that, if we were to fix the given depth, wo 
should thereby fix the given diameters; so, if we were to fix one 
of the given diameters, we should thereby fix the given depth and 
the other given diameter. But we cannot fix the depth of a flar- 
ing vessel by describing it upon the side ; nor can we in any way 
fix the given diameters, except by Uieir circumferences upon the arcs 
lohich we have drawn. 

It thus appears, that, before we can proceed to complete our pat- 
tern, we must know the circumference of one of the given bases, 
and must decide what the measuring unit of the pattern .^^hall l^; 
whether one covering the whole lateral surface of the vessel, or 
only a right section of that surface, covering a known aliquot part, 
as ii \^ i» or less, of that surface. 

Suppose, now, for example, that one of the given diameters of 
the pattern thus far drawn, we will say the greater, is 18 inches, 
and that we would take out a jiattern covering say one-fourth part 
of the lateral surface or side ; then 3.1410 X 18 = 56.5488 inches, 
the circumference of the greater given base ; and 5G.5488 -f-4 = 
14.137, or 14^ inches nearly, the length of the greater Jirc that 
will contain one-fourth part of the circumference. Then, with a 
strip of flexible tin cut to that length, and bent to the curve, or by 
any other mechanical means, measure off on the greater arc from 



PATTERN CUTTING. 331 

a, 111 the direction n, 141 inches large, as from atoh; and from the 
new-found pomt, b, draw a right line to the central point, i?, as 
bd R : then will the section a b d c he the pattern required. 
I his may be taken out with or without the requisite marf/ins for 
locks, burrs, &c., as desired, properly marked, and kept fo? future 
use. ^ 

in fw.lf^r\^^ ^^^^^ }.^ ^T.^^^ uncertainty and confusion, I shall in all cases 
i?Ai!i!^r> 1^ ^^°^^^^.*^^® directions to the construction of the nniENSioys- 
f^m«.^?;^!c''''^'°# *' n workman to allow the requisite margins for locks, 
f^nTrUn^ '' and rolled or wired rims, as his taste and the circumstances 
^^nJSVc.J'\ } ''''^^ ^"^'g^'f ' however, that, for a straight lock, the allowance 
W fhAh- \ ""^ on each side, and that two-thirds of the allowance on a side, 
if^ ^- ' ^'^^.^^'^i^ss of the pLate, should be turned when neatness and accuracy 
?uffi;w"?n?"fi ''? intended; also that, ordinarily, three-eighths of an inch is 
sumcient for the lock on tin-plate, and one inch for that on stove-pipe sheet- 

By Mathematics. 

R = radius, or slant height, of generating cone = E a, diagram. 

5 = slant height of given frustum = c a, diagram. 
i7=i perpendicular height of generating cone=:i? m, diagram, 
^^perpendicular depth of given frustum =:^ m, diagram. 
D — diameter of base of generating cone, or of greater base of 
fi"ustum. 
, d=z diameter of lesser base of frustum. 

r = radius, or slant height, of cone whose base is the lesser base 
of the given frustum = R c, diagram. 

Hd Rd ds , 

--HZIJ = ~ = J + d = 2^R^-H~2s/s^-h^ + d, 

(H — h )D Br ^ Ds ,- 



r = R — s=: 

S=zR — rz=:z 



rr nil Rh 



332 PATTERN CUTriXG. 

pROB. 2. — To construct a Pattern for the Body of a Vessel in the 
form (\f a Friistuyn of a Cone of f/iren Dl/nensions^ wiUiout plot" 
tiru) the dimensions^ and to take it out a knou^n measure of the 
lateral surface refjuired. 

KuLK. — Take the ra<lius of (he arc of tlie [rreatcr given base in 
the dividers, or beam compasses, and from any desired centre on the 

pLite, from which the pattern i» 
to !)c taken, as o, diagram, iK»- 
scribc an arc, as Jt t\ to any suf- 
ficient lenjith ; then with the ra- 
dius of the arc of the h'sser 
ffiven base in the dividers, or 
beam compasseg, describe, from 
the same centre, a paraUel arc, 
as /• n, to any siiHicient h'nj:th; 
then draw a ri;^ht hne from the 
central |K)int to tlie outer arc, 
as o r R^ diagram. Next apply 
the re(|ui>ite measure to the arc 
of the base whos<* circumfer- 
ence is known, as from Jt to t«, 
diagram; and fmm the point v draw a right line to the centraP 
point, as v n o: the sei tion It v n r will be the pattern proj>oscd. 

ExAMPi.K. — The depth of a vessel is to be 10 inches; the 
diameter of one of its bases, 8 inches; that of the other, 6 inch- 
es ; and a pattern containing one-half the lateral surface is re- 
quired. 

Now, by the foregoing formula;, R = ^~IV-{- {^D)\ and 

// = •=- ; therefore 

D — d 




^ = J {^~lj + (i^)'- '*» ^y ^^^ formula; = ~ ; then 



D 



I (- » > ) + (§)*= •^'^•- inches, the radius of the arc of the 

greater given liasc, and — "- z=. 30.10 inches, the radius of the 

arc of the lesser given base. 

8 X 3.141G-^ 2 = 12.5664 =::12i<v inches, short; the required 
length of the arc of the greater base ; or 3.1416 X 6 -f- 2 = 9y^ 
inches, large, the required length of the arc of the lesser givea 
base. 



PATTERN CCTTIXG. 



833 



Note.— As before gtated, it is wholly immaterial which of the arcs is 
known, for by defining one we define the other ; but generally the greater arc 
can be much more readily measured by nieclianical means than the h^sser, and 
it may be trimmed to facilitate the act, in which case the measuring-tape may 
be used. When the lesser arc is to be measured mechanically, tiie measure 
should be a strip ot flexible plate, cut to the required length, and bent to the 
curve. 

If it is desired to convert the decimal part of an inch into eighths of an inch, 
multiplv it by 8; if into sixteenths, multiply it by IC. Thus, the decimal 
.6604 X 's = 4.5:n2 eighths = 9.0024 sixteenths. The decimal 5G44, therefore, is 
practically equal to U-IC. 

PROB. 3. — To construct a Pattern for the Lateral Portion of a ves- 
sel in the form of a Frustum of a Cone, of given relative pro- 
portions or symmetry of outline, and r/iven Capacity ; any two of 
its dimensions, and one of them a hase^ being given. 
The following table has been calculated for relative propor- 
tions as set down at the top of the columns, and for portions, P*n, or 
parts, of lateral surface, as set down in the left-hand column. 

R represents the radius of the arc of the <ziven base, and D, 
the diameter of that base. Thus, if D be taken for the greater 
base, then R will represent the radius of the arc of that base, and 
the slant height of the cone from, which the frustum is to be taken ; 
but if D be taken for the lesser base, then R will represent the 
radius of the arc of the lesser base, and the slant height of the 
cone that will be left after the frustum has been taken from it. 

H represents the perpendicular height of the cone having D for 
its base. 

c represents the chord of the required arc, or chord that will 
subtend the arc that must be on the pattern or portion set down 
in the left-hand column. 

S represents the cubic contents of the cone having D for its base. 



P'n 


E = D. 


R = UD. 


R = W. 


R = HD. 


R = W. 


R = iD. 


R = GI). 


JiX=c 


RX=c 


RX=c 


RX=c 


RX=c 


RX=c 


RX=c 


1 


2 


1.8478 


1.4142 


1.1755 


1 


.7654 


.5176 


i 
i 


1.4142 


1.1113 


.7654 


.618 


.5176 


.39 


.2611 


1 


.7654 


.5176 


4153 


.3473 


.2611 


.1743 


d 
^ 


.7654 


.5806 


.39 


.3129 


.2611 


.1961 


.1308 


4 


.618 


.4669 


.3129 


.2506 


.2091 


.1569 


.1047 


1 


.5176 


.3D 


.2611 


.20.')1 


.1743 


.1308 


.0872 


6 


DX=H 


dx=b: 


DX=H 


DX=B 


DX=1I 


DX=B 


DX=II 


q 


.866 


1.23G 


1.9365 


2.4495 


2.958 


3.9686 


5.9791 




D^X=^S 


D^X = 'S 


D^X = S 


D^X=^S 


DoX=S 


D^X = S 


i)3 x = 'Sr 


h 


.2267 


.3236 


.507 


.6413 


.7744 


1.039 


1.5653 



When R = 5Z), and pattern covering the whole lateral portion 

I of the vessel is required, czn.Ql^R; covering 1^ c = .3129it; 
I-, cr=.2091i^; 1, C3=.1569i?; \,c = .l'2blR; i, czz: .1047i^; 



:4.975i); 



l, C3=.1569i? . 



S34 



PATTKKN CUTTING. 



llrLi: 



Gexkral Explaxation of the Table. — R = D ; then 
It X ^ is equal the chord of the arc that is equal to the whole cir- 
cumllrence of th»' hasc, and Ji X 1-4112 is etjual the chord of the 
arc that is equal to half the circumference of the base, &c. 

R z=z 2D: then R X .7Gi34 is equal the chord of the arc that is 
equal to half the circumference of the baseband Rx .5176 is 
equal the chord of the arc that is equal to onc-thiixl part of the 
circumference of the base, &c., 

'J'lie annexed <liaojram is drawn to radii 1, 1}, 2, 3, and 4 times 
the diameters of the ends, and exhibits symmetrical projwrtions 
for vessels of this general class ; and, by help of the preceding 
table, a pattern containinjj the whole lateral surface of either 
special fi;:ure, or such portion, thereof as set down in the Icfl-hand 
column, may be readily and correctly obtained. 

When the greater Base of the Vessel is given. 

Place the square suit^ible on the plate from which 
the pattern is to be taken, 
and scril)e to its blades, as 
a m, m R, Make m R oi 
sullicient lenprth,and make 
m a equal to any desired 
alif|uot part of the great- 
er given base ; then, with 
the s<iuare or straight-edge 
in the position a Ry and 
that measure 1, 1^, 2, or 
more times the diameter 
of the greater base, ac- 
cording^ to the table and 
the special fi^rure intend- 
ed, draw a line, or that 
measure, as a R. Next, 
if the perpendicular depth 
l>e given, space it oflT on 
the perpendicidar, as m h; 
and fromA.with the square 
as at first dropped down 
to that i>oint, draw a line, 
h c, parallel torn «, which 
will be the corresjKmding 
measure of the lesser base 
of the vessel. If the slant 
depth of the vessel, instead of the perpendicular, be given, space it 
off on the slant depth, as a c ; and from c, with the square as be- 




PATTERN CUTTING. 385 

fore, draw a line, c Ji^ the corresponding measure of the lesser 
base, as before. If the lesser base of the vessel, instead of the 
perpendicular depth, or slant depth, be given, take the same rela- 
tive measure of that on the square that was taken of the greater 
base ; and with the square as at first dropped dov/n on the perpen- 
dicular, until that measure is exactly included between the lines 
. a R and m it, as h c, draw a line, h c, which will be the required 
measure of the lesser base. Next, with the radius a it, and R the 
centre, describe an arc, as a 5, to any sufficient length ; and with 
the radius c R, and R the centre, describe a parallel arc, as c d, to 
any sufficient length. 

Suppose now, for example, that we have drawn the arcs to ra- 
dii once the diameters of the vessel ; that the diameter of the great- 
er base is 18 inches ; and that we would take out a pattern con- 
taining say I part of the lateral surface required : then, on turn- 
ing to the table, we find in the column headed it = i), and 
against ^ in the column of portions, the constant, or co-efficient, 
0.7654 ; and we are told at the top of the column, that the 
radius multiplied by that co-efficient is equal the chord of the re- 
quired arc. Then 18 X .7654 =: 13.779 inches, the chord of the 
arc that is equal to ^ part of the circumference of the greater 
base. Take, therefore, 13.779 inches in the dividers, and from the 
point a, with that measure, cut the arc, as at h, and from the new- 
found point, &, draw a right line to the point it, as 6 i^ ; then will 
the section a c d bhe the pattern sought. This may be taken out, 
and marked with the values of it, d, and ^, with other distinguishing 
marks, as the depth, or capacity, or both, and kept for future use. 

Suppose again, for further illustration of the practical use of the 

TABLE, that we have drawn the arcs to radii 1^ the diameters of 

the bases ; that the diameter of the greater base is 1 inches ; and 

j that we would take out a pattern containing ^ the side of the 

I" vessel : then, on referring to the column headed it = l^i>, in the 

[ table, we find in that column, against the portion ^ in the column 

of portions, the co-efficient 1.1113 ; and 10 X ii X*" 1-1113 = 14.82 

inches, the chord of the arc required. Take, therefore, 14.82 

I inches in the dividers, or on the square, and from the point a, with 

I that measure, cut the arc as at 6, and draw a line b R. The section 

a c db will be the pattern demanded. 

• 
When the Lesser Base of the Vessel is given, and the Greater is 

unknown. 

Rule. — Place the square suitably on the plate from which the 
pattern is to be taken, as c h R, diagram, and draw to any suffi- 
cient length a line, h R; and from h, on the other arm of the 



33G PATTERX CUTTING. 

squaro, set ofT any aliquot part of the base, as h r, and draw that 
measure. Next, with the square, or strai'jht edjje, in position, as 
c 71, an<l tliat measure K H, 2, or more times tlie diameter of the 
base, fiecordinjj; to the table and the speeial fi;iure intended, draw 
a line, as c //, and produee it suflieiently in the direetion a. Next 
pnxluee R h suflieiently in the direction in. Next, if the perpen- 
dicular depth is known, space it oflT on the perjKjndicular pro- 
duced, as li m, and, with the square raised tip to tho point ;/i, draw 
a line, in a, jiarallel to /< c, winch will be the cori' ' alicjuot 

part of the j^reater !)ase. If the slant depth is ica*! of 

the depth, space it off on the slant depth ])roducc<l, .is c a, and, 
with the s<piare raised up on tin* jH»rpen<lieular till the shorter arm 
cuts the point a, draw a line, a m, which will be the h^piired ali- 
quot part of tin* ;ireatcr base. Next, with It a the radius, and li 
the centre, desi-rilxj an are, as a h, to any suflicicnt hn,rth ; and 
with Jt c the radius, and R the centre, <leseril)e an underlying arc, 
as r J, to any sulHcient leuLjth. 

Siq)p<)se now, for example, that wc have drawn the arcs to radii 
six times the diameters of the bases, that the <liameter of the lesser 
base is 11 inches, and that we would takeout a pattern containing 
one-third part of the side. On referrin;^ to the table we find, m 
the column headed /t =fi7), and against } in the column of |x)rtion9, 
the constant .1743, and 11 Xf»X0.1 748 = 11^ inches, the chord of 
the arc of the lessi^r base that is equal to on<Mlnrd part of the circum- 
ference qf that base. Take, therefore, 11 J inches on the stiuare; 
and with that measure, as from c to r/, cut the an* as at fl; and from 
i?, throu'di the point J, draw a right line to the greater arc, as 
R (lb : then will the wclion a h d c l)e the pattern demanded. 

It may l)e proper to state that the table is also applicable in find- 
ing the sides or ends of a vess^d in the form of a prismoid, or of a 
frustum of a pyramid, of any numlxir of sides from thn»e up, cor- 
responding to the denominators of the fractional jwrtions tabulated. 

Prob. 4. — The special tabular f<jurp^ the Diameter of one end, and 
the Cubic Capacity of the vessel^ }>einfj givcn^ to find the Diameter 
of the other en<L 

Z), d = (lianu'tei*s ol' 1m-«^>. 

a = cubi(»capacii\ ni \,-sel. 
k = tabular constant for .S'. 

Example. — A vessel in tbo form of a frustum of a cone i^ to be 



PATTERN CUTTING. 337 

constructed to radii 4 times the diameters of the bases : the diame- 
ter of the greater base is to be 7 inches, and the capacity exactly 
1 gallon (231 cubic inches). What must be the diameter of the 
other base ? 

Seeking k in the column headed it = 4i), in the table, and under 
JD^X = ^, we find it to be 1.039 ; then 



^ 



7^X1.039 — 231 . .. , 

= 4.9416 inches. Ans. 

i.uoy 



Having now both diameters of the vessel, proceed for a pattern 
containing the desired portion of the side, as directed in the fore- 
going. 

Example. — A measure of the exact capacity of 2 gallons is to 
be constructed to radii 3 times the diameters of the bases : the 
diameter of the lesser base is to be 5^ inches. What must be the 
diameter of the greater base ? 

^', by the table, = .7 744 ; then 

|5.5'X .7744 -[-462 



^- 



„„_ =9.1377 inches. Ans. 

.7744 



Example. — What will be the depth of the last-mentioned vessel? 

Seeking q in the column headed R z= SD, in the table, under 

D X =://, we find it to be 2.958;.-. ^=9.1377 X 2.958 = 27.0293 

77r7 
inches, and h = J/ -=- = 2.958 (D — d) = 10.76 inches. ^4^5. 

Example. — A pan in the form of a truncated cone is to be 
constructed to radii once the diameters of the bases : the diameter 
of the greater base is to be 15 J inches, and the capacity 8 wine 
quarts (462 cubic inches). What must be the diameter of the 
other base ? 

The constant for the solidity, in the column headed Rz=D, is 
.2267, and 

|15.5^X. 2267 — 462 ,, . . . . 
2^67 ~ ^^ inches. Ans. 

Example. — What will be the slant depth, and what the perpen- 
dicular, of the aforementioned pan ? 

s=zR — 7', and Jiz= Hs~ R. But R in this case = Z>, and 
r=z Rd-^D=^d: therefore s = 3.5976 inches. Ans. 

i7, by the Table, ==Z)X .866 = 13.423, and 7i = 13.423 X 3.5976 
-f- 15.5 =.866 (D-^d) = 3.116 inches. Ans. 
29 



338 PATTERN CUTTINO. 

By ^Mathematics. 

Djdzzz diameters of bases. 

R, r = radii of arcs of bases. 

H= perpendicular hei;iht of generating cone. 

h =z perpendicular depth of vessel. 

S = cubic contents ot •renerating cone. 

a =z cubic capacity of vessel. 

12.S' 1 2 Da . r2a 









12 Z>' — f/' 12(D--(/) 

_ S(D'^(P) _ 7rII(ir^fr) _ 7Th(Dd + D' + d') 
°"" D" 12D 18 

For other forms of expression, and other applicable equations, 
sec p. 331. 

pRon. 5. — To construct a Pattern for the Lateral Portion of a ves- 
sel in the form of a Frustum of a Cone^ of given Tabular OuUine^ 
and given Dimensions, without plotting the dimcn^'non^. 

s = slant depth of vessel. 

f7 = tabular ratio of R to J)= 1, U, 2, .», c\« ., i^p iuic. 

a=z tabular ratio of 7/ to />, third line from bottom. 

Ic =z tabular constant for S, bottom line. 

?/ = tabular constant lor chortl of arc. 

Other symbols as in the foregoing. 

When both Bases of the Vessel are given, 

R = Dg, and r = dg; A = q(D — d)j s = R — r. 

When the Greater Base and the Depth are given. 



J 



PATTERN CUTTING. 339 

When the Greater Base and the Slant Depth are given^ 
R 1= Dg, and r=:R — 5, 

When the Lesser Base and the Depth are giver^, 
R = r^d2±h^ ^ j,^Rd^R^d_^±h^ 

dq ' ^' r g q 

When the Lesser Base and the Slant Depth are given^ 
R^=zr -}-s ; r::z=idg. 

Example. — Given the tabular outline R=z 2 D, the diameter 
of the lesser base d=z5, and the depth ^ = 8, to find the diame- 
ter of the greater base D, the radii of the arcs R, r, and the 
chord of the greater arc c, that will subtend ^ the circumference 
of the greater base. 

By the Table, when ^ = 2, and p = ^; that is, when R = 2 D, 
and a pattern containing one-half the lateral surface is required, 
^ = 1.9365, and ^=.7654; 

D = -—-=d.l3', and = 18.26 X .''654 = 13.976, Ans. 

Example. — Given D = 10, s = 4, ^ = 3, 7^ iz: J, to. find R, r, 
and c. 

By the Table, (7 zz: 2.958, and ?j = .3473 ; then 2^ = 10 X 3 = 30 ; 
r = 30 — 4=26; and c = 30 X -3473 = 10.419. Ans. 

When the tabular outline Rz=^ Dg, the assumed diameter, and 
the cubic capacity only, are given, the other diameter must be 
found, and by one of the following formulas, viz. : — 



.= \?i=i^ ill''--) 



Rule. — Take R in the dividers, or beam compasses, and from 
any suitable centre, 0, describe an arc, R v, to any sufficient length. 
Next, with r the radius, and the same centre, describe an arc par- 
allel to the former, r n, to any sufficient length ; then draw a right 
line from the central point to the outer arc, and cutting the arcs, 
as at r and R. 



340 



PATTEKN CUTTINO, 




Suppose, now, for example, that the arcs are drawn to raclli once 
and one-third the diameters of the bases ; that the radius of the 
greater arc is 16 inches ; and that you wish to take out a pattern 
containing one-fifth part of the side of the vessel. Keferrin;: to the 

table, column Jl = Ij A and 
against^ in the column of |X)r- 
tions, you find .4GG9; and 
16 X .4669 zr 7.4 7 inches, the 
chonl rc(|uired. Take therefore 
7.4 7 inches in the dividers or 
on the scjuare, and from the 
point/?, with that mea^^ure, cut 
the greater arc as at r, and from 
the new-found point, r, draw a 
right line to the central point 
o, and cutting the lesser arc, 
as at u : then will It v n r he 
the pattern sought. 

On the contrar)', suppose you have drawn the arcs to radii four 
times the dianuters of the bases, that the diamet^^r of the lesser base 
is 5 inches, and that a p«ittern containin'^ one-third part of the lat- 
eral portion of the vessel is reijuired : then, by the table, column 
R = 4D ; and a;jrainst J in the column of |>orlions, the constant, .261 1 
appears; and .0 X 4 X .2611 =:,'>.222 inches, the chonl of the arc 
of the lesser base that is equal to one-third part of the circumfer- 
ence of that base. Take, therefon\ 5.222 inches on the wjuarc, 
and trom the point r, with that mea.sure, cut the lesser arc, as at n ; 
and from the central ])oint n, through the point n, draw a right line 
to the greater arc, and cutting that arc, as at v : then will the sec- 
tion It r ;i tJ be a pattern for the vessel, and contain one-third part 
of the side. 

Prob. 6. — 7'he Capacity in gallons of a Vessel in the form of a 
Frustum of a Cone being given^andany two of its dimensions^ to 
find the other Dimension. 

It has been shown that — ^^ ^- — ^ — - = solidity, or capa- 

city, of a frustum of a cone, the solidity and the dimensions being 
in the same terms of measurement ; 



gcs-'f)-?- 



But since a gallon contains 231 cubic inches, and 



4 X 231 



: .0034, 



PATTERN CUTTING. 341 

u Mows that ■^^m^±±^±jQ-=.oonik(Da + iy + cP) 

= solidity, or capacity in gallons; the dimensions being in inches. 

Therefore, putting C to represent the capacity of the vessel in 

gallons, 



\()034(Dd + D^ -{^ d- 






— 1/ ^C _3i)^ D 



Example. — A measure in the form of a frustum of a cone is to 
be constructed to the exact capacity of 3 gallons ; the diameter of 
one of its bases is to be 11 inches, and the perpendicular depth 12 
inches. What must be the diameter of the other base ? 



J(- 



3X3 irx 3' 

12 X .0034 4 



■)-"= 



5.895 inches. Ans, 



Example. — A vessel in the form of a frustum of a cone is to be 
constructed to the capacity of 3 gallons ; the diameters are to be 11 
inches, and 8| inches. What must be the depth ? 

.0034(11 X 8^+ 11' + 8.75') = '"^^ '"^'^^^ ^"^- 

Example. — A measure in the form of a truncated cone is to be 
constructed to the capacity of ^ gallon ; the diameter of one end is 
to be 5 J inches, and that of the other 4 inches. What must be the 
depth ? 

g y JL 

^ = 6.464 inches. Ans. 



.0034(22 -J- 30.25 + 16)' 



Example. — A measure is to be constructed to the capacity of 
one wine quart ; the diameter of one of the bases is to be 4 inches, 
and the depth 5^ inches. What must be the diameter of the other 
base? 



Jc 



3Xi = -75 _ 3 X 4^' 
.0034 X 5.5 



2\ 4 

-j — - =S^Q inches. A71S. 



Example. — A measure of the capacity of one wine pint is to be 
constructed ; the depth is to be 4i inches, and one of the diameters 
is to be 3-| inches. What must be the other diameter ? 



J( 



2d* 



342 FATTKKX CUTTING, 

Example. — A nwasiirc is to lioM ^ wimr pint, andeacli base is 
to be 2 1 inches in diameter. What must \xi the depth ? 

. 3_><_tV.„__0<^5 ___ _.0625 

.0034 X 3(/-' — . 0034^/^ "~. 0034 X 2.?75- ^ ^•^^•' ^^'"^ ^"'• 



Prob. 7. — Jb c/)nstruct PatteiTiS for flaring oihiI Vesselt of differ- 
ent Eccentricities and given Dimensions, 

The solids here contemplated, and for which coverings are to he 
constructed, closHy resemble the Trust crms of elliptic cones ; but 
they are not identical with that figure, since an oval is made up of 
circle aiTS, while no part of an cllij)se whatever is stri<'tly the arc 
of a circle. 'Hie leadin<^ principle, however, tor this class of pat- 
terns, so far as repjards their sides, is expres{*ed by the act of an 
elliptic cone rotating on its side, on a plane, from the line where? 
the plane of the transverse axis of its base b at ri;:ht anjjhs to the 
plane on which it rotates, to the extcntof one n»vt>iution of the cone. 

J) represents the transverse diameter, and r/the conjnjrate, of the 
greater base ; IJ' the transversa, and (V the conjugate, of the lesser 

base ; h the perpendicular depth of the vessel : /f z= "rrit), the 

perpendicular height of generating cone ; 3f = ^ H^~-\^{\D)\ the 

maximum slant height of generating cone; ♦^= V^-^* -t"(l<^}% 
the minimum slant height of generating cone. 

No. 1. — The Lateral Portion, D : r/ : : D' ; (F. 

tf=F; taz=P;fa = R;fc=N; fv=S: fA=M; dia- 
gram. 

Rule. — Place the scjyare suitably on the plate from which the 
pattern is to be taken, and scribe to its edjjcs a /, t ]i^ making both 
lines of sufficient length ; and produce t It sufficiently in the direc- 

tion.. Make ^f=W'+~^^^^^ 

= 0.345092A and draw fa. Make i i = p-j-I^^^^ and 

I L^ 

draw I in parallel to t a. ^nxt, with cfz=i UV on one blade 

of the square, and the limit of that measure in the point/, the 
other blade cutting the point a, !\sfc a, draw/c, c a, and produce 



PATTLT.:; CUTTING}, 



343 



Make tR=F 



MN 



and draw a R. 



y c\x\ the direction A, 

Make 

B^a R) 

and draw m r, 
which will be par- 
allel to a /^ Then 
with « R m the 
dividers, and E. 
the centr-e, de- 
scribe a v^' and 
with ??e ?• in the 
dividers, and r the 
*'cntr<B, describe m 
t; also with a ^ 
the radius, and y' 
the centre, de- 
scribe a A ; and, 
with m z the ra- 
dius, and z the 
centre, describe m 
C Thus A a V k 
m C will be the 
unit measure of 
the pattern, and 
will contain one- 
fourth part of the 
side. 



Note. — It will be perceived that the section on the right of the per- 
pendicular V 72, diagram, is an exact duplicate of that on the left, only reversed ; 
thus A V B Dk C is a pattern for the vessel, and contains one-half its lateral 
surface. The last-mentioned pattern may be obtained by draft, by repeating 
the lines and iircs on the left; or it may be obtained by scribirig'to the unit 
measure in the two positions. In making up, the side should consist of a 
single piece when practicable, and the lock or seam should be un<ier the 
handle. 

To construct the Bases for No. 1. 

Rule. — Lay off A B, diagram, equal in length to the trans- 
verse diameter of the base, and divide it into three equal parts, e 
and r; also bisect it, and through the point of bisection, o, draw an 




244 



PATTERN' CrTTTSC. 



indefinite perpeDdknlar, a» C D* Make f />, e /j, caoTi cqnaT f n 

A e or B r : and Iron • 
the jx)ints p and */. 
ihroQgli the [HAnis * 
and r, draw right 
lines, as r; e Syff r /, 
p e m, p r n. Then 
with A e €T B y th« 
radius, and c and r 
the centre.**, deseribo 
m A Sj t B u : and 
with g s or p u the 
radius, an«l (j and p 
the centres, ckraeriUt 
« t and m fi. 




8^/ 



By MATHEMATICf*. 



i4 /?=/) = — :^-,. = 1.822781 
4 — v^8 



-■o=rf = LLzzl^ 



^ = 0.288675/); w( ::^/7 />=: 



_r 0. 7559881/) ; 

(2-v3)^ 



p.«r: 



»— 1 . — - 



= 0.08981 64 D; 
17 .f=^ rz=i7C= -;. 



A y z= e j^ = o r =: ^ \ s g t=^ 60^ ; ?7i c .^ =: 120^ 



iVo. 2. — TJie Lateral Portion. D\d\\U \^. 

D represents the transverse, and d the conjugate, diameters 
of the greater base ; D' the transverse, and d^ the conjugate diam- 
eter of the lesser base ; h the pcrix?ndicular depth of the vcj 

H- -»*' 



iam- J 
sseL^M 

ri__jy' ^'*c perpendicular height of the generating cone ; '^ ' 



.S=: ^W-\~ (^d )*, the minimum slant height of generating cono ; 

M=: yZ/i^-f" (J^)*» ^^^ maximum slant height of generating cone. 

tf=F; ta = P; fa = R; fc = N; fv=S; 
f A^=^ M ; diagram. 

Rule. — Place the square suitably on the plate from which the 
pattern is to be cut (a i /?, diagram), and draw lines a t, t R, each 



PATTERN CDTTING; 



345 



of sufficient length ; and produce R t sufficiently in the direction w 
Uaketf= Ih'4.^, and<a= 1^^^^'= 0.42163 7 A 



^>-+§ 



w 



25 



and draw/«. Mak<2 t i z=z h^ -L 



all<il to ^ <t. Next, with c /= 



n1' 

^ 20 



[ ?and draw 



e m par- 



•on oae edge of the square. 



the limit ©f that 
measure in the 
point /, the edvje 
of the other bhuie 
cuttinnr the point 
©, as /* c a, draw 
fc,c a^ and pit>- 
duce/c in the di 
rection A. Make 

and draw a ii. 
Make 

tr_ — -^, 

and draw m r, 
which will be par- 
allel to a R : then 
with a R the ra- 
dius, and R the 
centre, describe a 
V ; and with r m 
the radius and 
r the centre, de- 
scribe ?n k: also 
with a y the radi- 
us, and y the cen- 
tre, describe a A ; 
and with m z the 
radius, and z the 
centre, describe 
mC, Then wiil 
il a V ^^ m C be the unit measure of the lateral portion of the ves- 




346 



• PATTERN CUTTiyC?. 



«el, and contain one-fonrib part of tbat portion. Sec Note at &€ 
close of the directions for constructing tlie lateral portion of Na 

To €ons(rii<^i the Bases /or Xo, 2. 
Rule. — Make A ^ equal to the transverse diameter of the base, 

and dl\ i«le it into four 
equal \nrts, in r. o^ 
and r : an<l tluou^rb o 
draw a pcqiendicular, 
C JJ, of sulllciint 
lenf»tb. Make o f\ 
o D, i^acb etjual to ont- 
thinlof A />, andlnuu 
C and 1), ihroi|p:b e 
ami r, draw C e m, 
C r n, he 5, J) r L 
Then with /> C the 
radius, and I) and C 
ibe <oi>tn'j«, dest^ribe 
^ C ty and ?;i /) n ; aliO 
*'v ' ntros, describe 




Willi A e or B r the railin«, 
m A 8 and f /? w. 



and 



By Matiikmatio. 



2ff 



^ 78<^.7l — 1060.4. 

No. 3 — 1 Ac Lu'a-al Portion, D \ U \ \ tl \ </'. 
Synil)ols, same as for No. 1 and No. 2. 



RuLK. — Draw at.tR, at right angles to each other, and of suf- 
ficient length ; and produce R I sufficiently in the direction r. ISIako 

'/= l^^+^» «nJ /« = V/-1^07214Z>»= 0.3882285A and 



draw />. Make/ ? 



— j/i* + -~^—- L , and draw i m parallel (o 

a /. Next, with c /= //^^ on one blade of the square, and 

the limit of that measure in the point /, the other blade cutting 



PATTEHN CUTTIN<5. 



347 



The 



tlte point a, ^s fc <x, draw /c, c a; and produce / c sufficiently 
in the direction A. Make t Rz=.F -\- —77-' and draw ait. Make 

i r=: — J: — 1 and draw m r, which will be paraliei to a R. 

with o. R the radius, and R the centre, describe the arc a v; 
and with r mthe radius, and r the centre, describe the arc 7Ji k; 
also with a y the radius,- and y the centre, describe the arc a A ; 
and with m ;2 the radius, and z the centre, describe the arc 
m C: tlien will the section A a v k m C ha the unit measure of 
the pattern, and contain one-fourth part of the side. 

To construct tlve Bas<is for No. 3. 

Rule. — Make A B equal to the transverse diameter of the 
base, and divide it into four equal parts, in e, o, r ; then bisect it 
with a perpendicular of suf- 
ficient length. Make e ^, 
e ^, each equal to half yl B^ 
and from the points i and g^ 
through the points c and r, ' 
draw i e 7n, i r n, g e 5, 
gr t, all of sufficient length. 
Next, with A e or B r the 
radius, and e and r the cen- 
tres, describe the avcsmAs^ 
t B m; and with g s or g t 
the radius, and g and i the 
centres, describe the arcs 
s C tj m D n. 




AB = D = 



By Mathematics. 

^'^ ;=:l,57SUd; d = ^^ "/^^^ ^ ,6339746^- 



3— 'V/3" 



SD 



op 



_i>V/3 



z=:.21650635i>; p C =z^^-~J^^~ = Am^SlD; 

Ky ^V 3 

Ay ==.-—', 

i)v/3 , 3D 

4 4 *^ 



S48 rATTEKN cirrnxo. 

Hie Lateral Portion hy Mathematics, be the Eccentricity wJiOl \l may, 

p s:=:s, sine of the arc s C\ base ; 

op = {/, sine of the arc s A, base (sec diagram of base). 

D, Df, dy d\ h, Jl as in the forej,'oing, D'.D' \\d\ rf', 

m r = v/[(.- r)« + (i m)^ = ^'^^^> ; » m = v^[(m r)' - (.* r)^ 
= V/[(m/)^ - (ly)] = ^^; i r = v^[(m r)« - (i ;«/] = ^^^^ 

yiz = r' = Wr^D; xm=p' = iyp^D; xz=zf' = iyf^ D; 
ik = D'{S-^F)'^D;fi = F^(ti)\ Cx = D\M — N)^D;\ 
c A zz: M — iV; / 1; = S — F; also, I or A a r, diagram^ 

IT 

nearly ; and 4^ -f- tt =z diameter of generating circlo of cllipee. 






Of Cylindrical Elbows. 

Tlie solid to be taken into view in n^fcrence to the conjrtrnction 
of the arms of a hollow cylindriral elbow is a cylinder of the di- 
ameter proposed for the arms, havinu; one of its l)ases oljli<jue to 
the sides to the extent of half the angle projx)sed for the i'lbow; 
and the best mode of laying olV the arms, generally sj>eaking, is 
expressed by the act of this cylinder rotating on its side on a j)lane, 
from the line where the plane of the transverse axis of its oblique 
base is at right angles to the plane on which it rotates, to the ex- 
tent of one revolution of the cylinder. 

♦This formula for one-fourth of the porimetcrof an eliip**!- mi.»nl9 almost 
strict accuracy when the conjugate diameter is not greater than two-third:*, nor 
less than one third, of the transverse. It furnishes it too short by 1-1400 
when the conjugate is ecjual to three-Iourths of the truu^sverse, and too long^ 
by MlOO when the conjugate is cquai to onc-louxtli of ilie tranftverae. iSee 
Conic Sections, Ellipse, 



r'ATTEPtN CUTTING, 



349 



It is apparent, then, that the true face, or outHne, of the joint 
of a cylindrical elbow, when the cylinder in continuance is rolled 
into place, is an ellipse, whatever be the angle of the elbow. And 
it may be shown that if a hollow cylinder having an oblique base 
be bisected by the plane of either axis of that base, and the semi- 
cylinders opened to plane surfaces, the portion of the curve that 
will be on each will be a hyperbola, or two equal and similar semi- 
hyperbolas united, an-d forming a cima, or facing in opposite direc- 
tions, according as the plane of the conjugate axis, or that of the 
transvei-se, be made the cutting plane alluded to ; and it is clear 
that the curve that will be on one piece will be equal and similar 
to that on the other in all particulars. Thus the curve of the 
joint of a cylindrical elbow, when laid o\Y on a plane, is made up 
of four equal and similar semi-hy[>erbolas for either arm of the 
elbow ; the method of locking the joint not now being taken into 
account. 

From the foregoing analysis, the following rules and directions 
for practice have been dei'ived ; and it is believed they will be found 
not only as correct and simple, but as ready of execution, as any 
that have been or can well be devised for the purposes proposed. 
Prob. 8. — For a Right-angled Elbow. ^^^ 
(/=: diameter of pipe, 
d X .7854 = i circumference of pipe. 

Rule, — Construct on any plate suitable for the purpose, and 
for taking out, a rectangle, A C B D^ in length, A D, equal to 
one-fourth part of the cir- 
cumference of the pipe, and 
in breadth, A C, equal to 
one-half the diameter of the 
pipe. Make A <? equal to 
f/X 0.5708, and divide the 
space into any number of 
equal parts. Divide C B 
and B D each into the same ' 
number of equal parts that 
you divide A e into. Con- 
nect the points of division in 
A e and C B directly, by right lines, as 11, 2 2, 3 3, kn. Then, 
from the points of division in B i), draw right lines tending direct- 
ly to the point A, as IJL, 2-4, 3 .4, &c. The intersections of the 
lines bearing like numbers will be so many points in the locus 
(place or line) of the required curve ; and the more numerous 
these, the more completely, of course, will the curve be defined. 
The practice may be called locating the curve by intersecting lines, 
30 




350 



PATTERN CUTTING. 



and is equivalent to locating it by onlinatcs. Trace the corre, and 
the unit measure, A B C\ of the true arm of the contemplated eW 
bow will be constructed, and may be taken out for um* 

PROB. d. — To apply the Unit Measure to the Construction of a 
Full Pattern for the Anns of the Elbow, 

Rule. — Draw a ;ruide-lin(», C a (\ .«»uitaMy on the plate from 
which the pattern is to be taken, and of sutruient len^rth. Then, 
with the measure in position 1, scribe to 
its sides (^ .1, -1 B; witli it in jMjsition 2, 
scrifx? to its sides B Ay A a; witli it in 
position 8, scribe to its carve A B ; and 
with it in position 4, scribe to iti side» 
B A, A (\ This practice, care iK'inp taken 
during the pmceodings to keep the base 
of the measure diret tly in the gui<k»-Iinc 
C a (\ and that the extremities of the 
curve are made to unite in the »amo 
poinL**, will construct tlie curve A A Af 
the curve proposed. 

Now, a.^ may be remlilv inferred, were 
the workman to cut out botli arms of tho 
elbow by this curve, and t)ien to roll them 
pro|>erly into cylinders, they would nnite 
by their oblicpie bases unil'onnly through- 
out ; and w«)uM form «'* right angle, or 
stand to each other at an angle of 1)0 de- 
grees. Moreover, were he to turn a burr 
on tho ol)li(|ue base of one of them, and 
a ledge and lip to mateh on that of the 
Other, both parallel in their pra<ti(al bear- 
ings to the plane that is common to lx)th 
the said basses, the same state of things 
"would still be maintaino<l ; and the arms would lock with a cloeo 
uniform flange bisecting the angle. 

We have thus far spoken of the true or geometrieal cylindrical 
elbow ; but wo come now to sjx»ak of the arms as they orrlinarily 
come from the edging machine^ which, hhort of the strictest han- 
dling with reference thereto, can scarcely be made to turn the wards 
as above proposed, and, moreover, the flexibility of the plate is 
usually insuflicient to admit of it. 

The work, however, as it onlinarily comes from the edging- 
machine, lacks uniformity and definiteness in nearly all particulars. 
The wards, beside being out of parallelism to the plane that is 
common to the oblique bases, arc more or less irregular, and out of 




PATTERN CUTTING. 



351 



harmony with each other. Scarce any two workmen turn them 
alike ; and scarce any workman, it may almost or quite be said, 
ever turns them twice alike. The condition of the machine at one 
time compared with another ; the thickness of the plate employed ; 
the customary manner of handling the work in the machine ; and 
the fact that one of the arms has usually to be made less in circum- 
ference at its extremity than at the joint, that it may enter the 
flush end of a straight pipe of the same nominal diameter, which 
tends to displace the curve on that arm, — these circumstances, I say, 
render the irregularity and indefinitcness spoken of unavoidable, 
in a greater or less degree. 

There is one point, however, which we can fix upon, viz., — the 
wards come from the edging-machine invariably less oblique in 
their practical bearings to the sides of the i 
cylinders than comports .with the angle 
proposed for the elbow, unless that angle 
is considerably greater than 90 degrees. 

It is the practice with many workmen 
to retain the true curve for the outside arm 
of the elbow, and after closing the straight 
lock to turn its Avard unhesitatingly upon 
it ; and then, after closing the straight lock | 
of the inside arm, and before turning the 
burr, to cut away or trim its upper limb, 
or crown, until it will properly enter and I 
lock. But this practice does not tend to 
correct the angle ; and unless the wards 
are turned well down in the throat, and 
rather narrow at the crown, the angle will | 
be greater than demanded. 

The best general rule that I have been I 
able to put to a practical test for construct- 
ing a general pattern for both arms of the 
elbow, of from 4 to 8 inches in diameter, | 
is the following : — 

Rule. — Construct the curve ^ ^ ^, I 
with its lateral and central guides, as al- 
ready directed ; then with the measure ' 
successively in positions 1, 2, 3, 4, but dropped down equally in 
each position, according to the desired width of the pattern, con- 
struct the curve a a a; taking care the while to keep the perpen- 
dicular of the measure directly in the guide-lines, C A, produced; 
and that the extremities of the measure unite in the same points. 
This Avill duplicate the true curve, and place the curves parallel in 
position. At this stage, it will be well to make the central line 




352 



PATTERN CUTTING. 



A a flistinct and permanent and to drop the permanent perpcn- 
dinilars B b, B h. Next, make A in, A //, each criual to one-lburtli 
of an mch, and make a i, a c, each etjual to one-eighth ol' an inch. 
Then, with the measure nearly in position 1, its auirle B in the 
point 7i, and its curve tendin^: to the point m, scribe to it;^ curve, as 
B m ; and with its anprle B m the point h, and its curve cuttin^r 
the pomt /, scribe to it^ curve t h; also, with the measure nearly in 
position 4, its annrlo B in the point B, and im curve tcndin^r to 
the point n, scribe to its curve, as B n ; and, with its angle B in 
the point h, and its curve cutting the point e, scribe to its curve c b: 
' then m A n e (i i will be the pattern contemplated. 

ri:oB. lU. — To apply the Pattern to the CnnMruclion of the Amu 
of the Elboic, Pkob. 9. 
KuLi:.— Place the pattern suitably on the plate from which the 
arms are to be taken, and scribe to its curve m A n ; then raise it 
up on the plate till the points h b are in the points B B, and scribo 
to the curves i b, e b : then the curve tn A n, with the continuation 
of the plate, will be the outside arm of the elbow, and the curve 
t A e, with the continuation of the plate, will be the inside arm. 

Pkob. li.— To find A C of the Unit Measure for any Angle of 
Ell)0w whafcrer. 
Rui.K. — Draw three sides of a parallelogram, as A C, A D, 
D /;, diagram. Make .1 /^^ecpial to half the diameter of the arms', 
and make .1 (\ 1) B, each ofsudicient length; then, with the bevel 
s<|iiare set to half the angle pro|X)sed for the cIIkjw, as C A B, and 
one of its arms directly in A L\ scr-l*" t- '^^- "-"^ ! B : then will 
D B be equal to A C required. 

By Mathematics. 

For the Unit Measure of an Elbow of any r/ircn Angle. 

f/;i= diameter of arms = conjugate axis of liyperbola. 
A D=:a iizzd-^A =z .7851r/zz:^ circumference of arms. 

V= angle of elbow in degrees. 
A e = 2a — (l=i:^-(l — (l=:z,d70Sd. 

A C = b=: ^'^ __ idXcosiV 
tan^r~ s'ln^V 

db . 

t = transverse axis of hyperbola = -5-G</ + y^a*-|- ^d*) 

X =z abscissa from vertex = -j^dd' -j-if) — J/. 
y = ordinate = - \/(tx -j- xr). 



PATTERN CUTTING. 



353 



pROB. 12. — To construct- a Right-angled Elliptical Elbow. 

Rule. — Construct a rectangle, A D B C,'in length, A Z>, equal 
to one-fourth part of the circumference of the ellipse, or elliptic 
collar, and in breadth, A C, 
equal to half the conjugate 
diameter of the ellipse. 
Make A e= 0.18169 multi- 
plied by the circumference 
of the ellipse, and in all oth- 
er respects construct the 
unit measure by rule, 
Prob. 8. 

Next construct the full 
curve, A A A, by rule, 
Prob. 9, and take out both 
arms by that curve ; then lock the arms by their straight locks, 
turn the proper ward on the outside one, and trim the inside one, 
if necessary, to match. 

Prob. 13. — To Jind A C of ilie Unit Measure of an Elliptic 
Elbow of any given A ngle. 

Rule. — Make A D equal to half the conjugate diameter of the 
ellipse, and in all other respects proceed by rule, Prob. 11. 




By Mathematics. 
For the Unit Measure of an Elliptic Elbow of any given Angle. 

C = circumference of ellipse. 
Fziz angle of elbow in degrees. 
c zzz conjugate diameter of ellipse. 
d= C~7r:= conjugate axis of hyperbola. 
Ae = ^C('K — 2)^TTz=z^nd — d. 
A D=:a = ^ circumference of ellipse. 



A C=b=z 



ic 



^c X cos J F 



db 



tan^F 



sin^F 



t = —i [i^+ v^(a^+ 4^0]=*^^2insverse axis of hyperbola. 



a? = J \^(id ^-\-y^) — it=z abscissa from vertex. 

d 
y=^j ^(tx-\-x^)= ordinate. 

30* 



354 



PATTERN CUTTING. 



Prob. 14. — To construct the Quadrant of a given Circle 1/y inter- 
secting lines, 

IluLE. — Construct a sfjiiarc, A B C D, making each side equal 
to tlio radius of tlu* pro[K)so<l circle, an<l make A e e<|ual to the di- 
ameter of tlic circl'- ; tlicnci- bv rule. Prob. 8. 



1 



riiou. 15. — 7V> con.<trnr' tiir (lua(ira)tt oj a gircn JiUIjjsc hy in- 
tersecting lines. 

lluLK. — Constru<:t 
areetanglc-l BCD, 
makinjr A B ecjual to 
half the transverse 
diameter, and A D 
cc|ual to half the eon« 
ju;;ate. Make A e 
e(|ual to the oonju- 
jzale ; thence by rule, 
Prob. 8. 




Prob. 1G. — To apply the Quadrant of a Circle, or Quadrant tfrni 
Ellipse, to the Construction of the Circle or Ellipse. 

UvLK. — Draw a *rui(lc-liiu' of suflicientlcn;zth ; then, with A B 
of the ([uadrant in that line, in the four ivcjuisite positions, scribe 
to the curve and to Z^ C, as required. 

Prob. 17. — To construct the Quadrant of a Cycloidal Ellipse by 
intersecting lines. 

Hulk. — Construct a rectangle, ABC J), makinjr the trans- 
verse diameter to the conjugate as tt to 2 ; in all other resj>ects 
proceed by rule, Prob. 15. 

l*ROiJ. 18. — To describe an Ellipse of given diamrfers. h>i means 
of iioo Posts, a Pencil, and a String. 

I^et A B be the transverse diameter, and C D the conjugate. 
Rule. — Lay down the given diameters at right angles to each 




PATTERN CUTTING. 855 

Other, and bisecting each other, as in o. Next, with half ^ ^ in 
the dividers, and one 
foot in C or D, cut 
A B'my and z. Next, 
strike in a pin at // and 
another at z. Next, 
pass a string around 
the pins, and tie it at 
such a length that the 
loop may be extended 
to C or D. Next, In- 
troduce a pencil, and, 
bearing upon the 
string, carry it ai-ound 
the centre, to the con- 
struction of the full 
circumference. 

Note. — If we let I represent the circa inference of an ellipse, D the trans- 
verse diameter, and d tlie conjugate, then Z = ;'.53 -y (~ t— H ~ — i* 

very nearly. And I ntay add, that the expression affords almost strict accuracy 
when d is not greater than two-thirds nor less than one-third of T). It gives'/ 
too long by 1-274: wlien d=:}D, and too short by l-3.jl when d::zlD, (See 
Conic sections, Ellipse.) 

Prob. 19. — To construct a Seml-pardbola hj ini er^ecting lines. 

KuLE. — Construct a rectangle, A B C Z>, making A B ecjual 
tj) the altitude of the parabola, and B C ec[ual to half the base ; 
make A e^=zB CzizlA D ; thence by rule, Prob. 8. 

Prob. 20. — To construct a Right-angled CircuJarr-Elhow of S, 4, 
5, 6, 7, or ^ pieces^ of any given Diameter^ and any given Radius 
of Curve. 

d = diameter of pipe. 

7' =r radius of throat (usually = ^tZ, f J, ^d, or d). 
TTc? -^ 4 = .7854 J = \ circumference of pipe. 
V=z angle at centre in degrees. 



The Unit Measure, 
d X tan V 



A D=zC B=z.7SbAd, 
A e = .b70Sd. 



356 



PATTBRN CCTTINO. 



No. of 


Anple at 


Aa = 


.^* = ,r 


bass 


A C« 


pieces. 


Centre. 


(</-t-r)X 


(f X tan r 


rx 


dxhtan V 


3 


22- 30' 


.41421 


.41421 


.41421 


.207105 


4 


15^ 


.26795 


.26795 


.26795 


.138976 


5 


11° 15' 


.19891 


.19891 


.19891 


.099455 


6 


9<> 


.15838 


.15838 


.15838 


.079190 


7 


7° 30' 


.13165 


.13165 


.13165 


.065825 


8 


6° 25' 43" 


.11266 


.11266 


.11266 


.056331 



1 




Rule. — Construct a rectangle, A C B D, in length, A D, equal 

to one-foui-th part of tlie cirfumforcnce of the pipe, and in width, 

A (\ ocjual to ^fl X tan V. 
Make A e = (lX 0.5708, and 
in all other res|>cct5 proceed 
by rule Prob. 8. 

Sui)[)ose the unit measure 
of a rijiht-angled circular el- 
l>ow of 3 pieces i.s nnjuired, 
and that the diameter of the 
pipe is to Ix? 7 inches; then, 
on turning to the table, col- 
umn hraded 

A C=(lX itan V, 

and opposite 8 in the column containing the number of piece.*?, we 

find the co-eflicient, or half the tangent of 22° 30' to be 0.207105; 

therefore A C of the unit measure = 7 X 0.207105 = 1.449735 in.; 

A /)= 0.7854 X 7 = 5.4978 in.; and i4e = 0.5708 X 7 = 3.9936 

inches. 

Suppose the unit measure of an cll)ow of 5 pieces is required, 

and tliat the diameter of the pipe is to be 8 inches, then 

A D = .7854 X 8 = 6.2832 inches ; 
A C = 8 X .099455 = .79564 inch; 
A e = SX .5708 = 4.5664 inches. 

On the contrary-, suppose the unit measure of an elbow of 4 
pieces is required, and that the diameter of the pipe is to be 
6 inches, then ^ 

A 7) = .7854 X 6=4.7124 inches; 

^ C = .133975 X 6 = .80385 inch ; 

il e = .5708 X 6 = 3.4248 inches. 



PATTERN CUTTING. 



357 



PROB. 21. 



• To apply the Unit Measure to the Construction of the 
several Segments of a Circular Elbow. 

Rule — Draw a guide-line C C^ equa) in length to the circum- 
ference of the p.pe, and construct the curve, h .iV, bv scribin<.To 
the measure m its several positions, 1, 2, 3, 4, as by rule, Prob. 9 
Ihis will make A C =^ C h~ ^<] v ran F /,..! J-n i 
r X ^Tn 1/ ^^)=/^X tan F. Make h a, ^' «', each equal to 
1 X tan F, and draw the ime a «', which will make .1 a .V a' 
each equal to {d + r) X tan V. Thus /. yl Z.' a' a will be the unit 
T^T'U,^" '^^""V r^''^^'^^^ ^^^ ^'^^^1"^'^^^^ continuation oiTe 

pn?.l, w''' inside segments are equal one with anolher, and 

each IS equal to two of the outside segments, in all cases 
thnfT'^'r'' ^'-^^^-f^^e^i .^'i^^:"lar elbow of 3 pieces is required: 
thaf the diameter of the pipe is to be 7 inches, and that the throw 
or radius of the throat, r, is to be 6 inches ; then 

C a ova a' of the segments 3. 14 IG X 7 = 21.9912 inches. 

^ 6 = C & = ./ Xi tan 22° 30' =. 7 X .207105=. 1.449735 in 
^^ « = r X tan 22° 30/ == 6 X .41421 ::=^2.48526 in. 
Aa=(^d + r) tan 22° 30'= (7 + G) X .41421 =5.38473 in. 

The middle segment, therefore, of a 3-piecc cir- 
cular elbow will take the ibrm No. 2, or No. 3 
following diagram (No. 2, with reference to econ- 
omy of stock, when the outside sc<rments are 
made to lock at the throat). Thus the maximum 
width of the middle piece of a circular elbow con- 
sisting of 3 pieces = 2 {A a) = 2(d -f ,•) tan 22° 
30 ; and the mmimum width of the same piece 
= 2(b a) =2 2r X tan 22° 30^ 

Suppose a 5-piece elbow is required ; that the 
diameter of the pipe is to be G inches, and that 
the throw, or radius of the throat, is to be equal 
to f the diameter, or 4 inches ; then 

. C^C/ = 3.1416 X 6==] 8.849G inches; 

^C'=CZ/ HZ .099455 X 6 = .59673 inch: 
ba = .19891 X 4 = .79564 inch ; 

Aa=(6-\. 4) X .19891 = 1.9891 inches. 
^ On the contrary, suppose that a 6-pIece elbow 
is required ; that the diameter of the pipe is to be 
8 inches, and that the radius of the throat is to 
be equal to ^d, or 4 inches ; then 

.1. ^. ?i= ^'^^^^ X ^ = 2^-1^28 inches ; and, by 
the table, ' ' ^ 




358 



PATTERN CUTTING. 



A C= C6 = .07919 X 8 = .63352 inch; 
h a — .15838 X 4 = .G3352 inch ; 
A a = .15838 X (8 -|- 4) = 1.90056 inches. 
From the foregoing it may be perceived that to cut stock with 

reference to economy, 
we must make use of 
the three annexed gen- 
eral forms for the seg- 
ments of a circular 
elbow whenever the 
elbow is to consist of 
more than three pieces ; 
whereby one or more 
of the segments , will 
have its straight lock at 
the crown, and the oth- j 
cr, or others, at the | 
throat ; and, thus, an ' 
cll)Ow consisting of an 
o<ld number of pieces 
will make up without 
waste. 

The two outside seg- j 
mcnts of a circular* 
ell>ow arc e(]ual and 1 
similar, one to the other, I 
in all cases; and, if 
they are to have their | 
straight locks at the I 
throat, arc but exact 
copies of the unit sognient above «les(Tibc<l; thus, the segment 
h A b' a' a, preceding diagram, is identically the same as seg- 
ment No. 1, of the diagram here presented. It may be perceived, 
also, that the half of segment No. 1, by its dotted transverse, is 
identically the same as one-fourth of segment No. 2, by its dotted 
transverse and longitudinal sections ; and that a right quarter sec- 
tion of No. 2 is identically the same as a right quarter section of 
No. 3, only reversed in position. 

Note. — The half of segment No. 2 or 3 by tlie dotted cross-section is a 
good working measure for the segments, and it may be taken out bearing the 
requisite margins for locks or laps, both along the curves and at tho ward 
end. The allowance along the curve on one side should be with rcf< "> 

the {mn\ and, on the other, to the let/rfe and ///). One of the piec» 
the continuation for the pipe (one of the outside pieces) should be ii 
and trimmed to correct the angle, if necessary. But, with proper allow aiioc 
for locks, and correct handling in the edging-machine, trimming will eeldom 
be required. 




PATTERN CUTTING. 



359 




The annexed diagram presents a side view of a 5-segment elbow, 
or, in other words, 
it shows the outline 
of the several seg- 
ments when they 
are rolled into place. 
d represents the di- 
ameter of the pipe, 
and r the radius of 
the throat. The 
dotted arcs have 
their centres at o. 
In this diagram, r is 
taken equal to ^d. 
and it should seldom 
or never be taken 
at less ; in practice 
it may be taken to 
any extent greater, 
a s circumstances 
may require. A 
3-piece elbow is by no means a handsome structure, but it is easily 
made. A 4-piece elbow looks well, though it has one of the angles 
of its perimeter vertical to the centre. 

Prob. 22. — To construct a Collar for a Cylindrical Pipe of the 
same Diameter as the Receiving-pipe, 
The unit measure of this description of collar has already been 
treated of, and the manner of constructing it explained. It is 
identically the same as that of a right-angled cylindrical elbow of 
the same diameter. Directions for its construction are given under 
Prob. 8. 

Prob. 23. — To apply the Unit Measure to' the Construction of the 
Collar, Prob. 22. 

Rule. — Meas- i 
ure down from the 
top of the plate 
fi^om which the 
collar is to be tak- 
en, equal to the 
intended length of 
the collar and half 
its diameter, and 
draw a horizontal ' 
guide-line, B B B; 




next with the 



of the 



360 



PATTERN CUTTING. 



directly in the guide-line, scribe to its curve and its perpendicular 
in the several positions, 1, 2, 3, 4 ; then, B A B A By with the 
continuation for the pipe above, will be the collar proposed. 

Nf>TK. — This collar will lock at one of its aneles ; and, ordinarily, this is 
the best practico. If it btj desired to lock the collar from the crown of one of 
the arches, countruct it by positions 2,3, 4, and place podtiou 1 in continuatiou 
of position 4. 

Piton. 24. — To construct a Cj/lindrical Collar of a given Diameter 
to Jit a Receiving-pipe or Cylinder of a greater given Diameter, 

Rule. — Construct a rectangle, A C B D, in length, A Z), 

equal to :f part of the circum- 
Terence of* thn collar ; and in 
breadth, .1 C, equal to } the 
diameter of the collar multi- 
plied by the diameter of the 

I collar and divided by the di- 
ameter of the receivinji-pipe : 
make A e e(]ual to the diam- 
eter of the collar multii)lied 

I by 0.5708. Proceed in all 
otlicr respects for the unit 

' measure by rule, Prob. 8. 

Pbob. 25. — To apply the Unit Measure to the Construction of the 
Collar, Pnon. 21. 




Rule. — Proceed in all respects by ruh* 



23. 



By Mathematics. 



For the Unit Measure of a Rif/ht Cj/lindriral Collar of any given 
Diameter to fit a Cylinder of a given Diameter, 

(/ = diameter of collar = conjugate axis of hyperbola. 
/> = diameter of receiving-pipe or cylinder, 
C B = y = .7854^/ = \ circ. of collar = base of measure. 
A C=^>v=z(P-^2D z= perpendicular of measure. 
A c = h:Td — d = .570Sd. 

(fx 

i == — 2- (W+ Va^y + y*) = transverse axis of hyperbola. 

y = ^- ^(txJ + .r'^); X' = ^5 v/((i./)« + yO- i^• ^ being any 

abscissa or part of x, reckoned from the origin A^ and y' the ordi- 
nate to abscissa a/. 



PATTERN CUTTING. 



361 



pROB. 26. — To construct a Cylindrical Collar to fit an Elliptic- 
cylinder at either right section of the Ellipse, 

KuLE. — Find, by any mechanical means, the radius of a circle 

arc that coincides best with the arc of the ellipse to be covered by 

the collar, and take twice that radius for the practical diameter, 7), 

of the receiving pipe. Then proceed strictly by rule, Prob. 23, 

for the unit measure. Apply the measure to the construction of 

the collar by rule, Prob. 23. 

Note. — It is not in keeping with the laws of geometry to suppose that an 
ellipse and a circle can have a portion of their perimeters common to both ; 
but it is customary with sheet-iron workers to lix a cylinrlrical collar to an 
elliptic stove ; and by proceeding by the foregoing rule, a collar will be obtained 
which will practically fit without trimming, unless the diameter of the collar 
is unusually large for the size of the stove. 

Remark. — Section B A B o B of diai^ram, Pnob. 22, shows the 
opening for the reception of the collar (or to be circumscribed by the 
collar) as it appears upon a plane surface. It may be formed upon 
a plane surface by scribing to the respective unit measure in the 
four positions indicated, in all cases. 

Prob. 27. — To construct a Cylindrical Collar of a given Diameter, 
to fit a Cylinder of the same Diameter^ at any given Angle to the 
side of the Cylinder. 

(i^r diameter of collar, F== angle of collar to side of cylinder. 
Rule. — Construct a rectangle, A C B 7), making A D equal 
to half the 
eircumfe r- 
ence of the 
collar, and 
A C equal to 
Ar/-^tan^F. 
Bisect^ Din 
E and drop 
the perpen- 
dicular E F, 
w^hich will di- 
vide the rect- 
angle A C 
B D into two 
equal and 
similar rect- 
angles. Make 
B n equal to 
^dXtan^V, 
and draw the line n o parallel to B F. Next, make A e and n e 

31 




';^ 



362 PATTERN CUTTING. 

each equal to 0.5708r/, and proceed for the curves in all re^peots 
by rule, Prob. 8 : thus, A F n D will be the unit measure of the 
collar, and with the requisite continuation of the plate above A Dy 
for the pipe, together with the necessary margins for the rivets and 
straight lock, will be equal to one-half of it. 

ExAMPLK. — A cylindrical collar of 4 J inches in diameter is to 
be constructed to fit a cylinder of the same diameter at an angle 
of 35° to the side of the cylinder ; then 

A 7), diagram, = 7rr/-i- 2 = 3.141G X 4.5 -1-2 = 7.068G inches; 
and, by the table of natural sines, cosines, and tangents of dilferent 
angles, it is found that the tangent of half the angle of 35* 
(tan 17° 30') is .31530 ; therefore, 

A C, diagram, = 4^/ -^ tan J r= 2.25-^.3 153 = 7.13600 in., and 
D n, diagram, = i(/ X tan jr=2.25 X .3153= .701)425 iu. 



pRon. 28. — To construct a Ct/Undrical Collar^ or Spout ^ of a ffiven 
Diameter^ to Jit a Ct/linder of a tjreater given Diameter at a 
(jiven Angle to the side of the Cylinder, 

d= diameter of collar. 
/)=: diameter of cylinder. 
V = proposed angle. 

Rri.r:. — ]Makc A D equal to Tf/«-|-2Z),and in all other respects 
proceed strictly by rule, Prob. 27. 

NOTF. — Dinfn'am, Prob. 27, irprcwnt* a right spmi-fiectioii of a collar, to fit 
a cylinder of the name diamoter n« the collar, ot nn angle of 45* to the side of 
the cylinder. It is a general guide for obli(iue coliart*. 



PATTERN CUTTING. 



363 



OF SPOUTS, 



Spouts for vessels are usually made conical, more or less, 
according to the stake on which they are turned ; and, hy common 
consent, are divided into 
two classes; viz., ''^tea- 
kettle spouts" to fit cylin- 
drical vessels; and " cof- 
feepot spouts '" or *' teapot 
spouts,'" to fit flaring ves- 
sels. 

In practice, no definite 
geometrical relations be- 
tween the spout and the 
vessel it is intended for 
are sought to be main- 
tained. Thus, the diam- 
eter of the spout relative 
to that of the body, its 
length, fiare, angle of 
inclination to the body, 
and place of attachment, 
are matters of taste or 
convenience, or both, 
with the workman who 
constructs them. Never- 
theless, the ideas of s}'m- 
etry and practical utility 
are not to be outraged, 
but, on the contrary, 
should be kept in view. 

Rules, strictly geomet- 
rical, might be given, 
covering probable cases ; 
but the workman, with a 
little practice, can much 
sooner design a becoming spout, and fit it satisfactorily by " trial 
and trimming," than in any other way. 

The annexed diagram represents the general outline of spouts. 
It is given in comparison with the true arm of a right cylindricail 
elbow, which it in a considerable degree resembles, that it may the 
more readily be understood. The full-lined figure is that of the 
spout ; that by the dotted lines, the arm of the elbow. 




364 



PATTERN CUTTING, 



Of Pitched or Bevelled Coveks. 

Prob. 29. — To construct a Bevelled Circle^ or Circtdar Caver, of 
a f/iven Rise and given Diameter. 

h = rise, or perpendicular height. 
c/ = initial diameter, or diameter of hoop. 
8 = slant hei;iht, or radius of ehicf arc, = o A^ tllaijmm. 
mr=sum of the widths of the burr and cdcje (alv)ut | inch), 

= A />', diagram. 
6- = \^((iriy 4- h^) ; R=zs + m, =z o /i, diagram. 

Rule. — From a common centre, o, properly chosen on the 
plate, descnbe two circles, taking U for the radius of one of them, 

and s for that of the other. 
Ivext, draw a radius from the 
centre to the outer cin le, as 
D^ diagram, and cat out that 
circle. Next, cut a narrow 
flexible measure, in length 
equal to iT(2/t'-<n, and from 
the ])oiut />, with tliat measure 
bent to the circumference, 
measure off Its lengrti on the 
circumfiTcnee, as from B to /**, 
and from tlie new-found point, 
F^ draw a radius to the centre, 
as o F. Next, bisect the arc 
B F, and draw the bisecting 
radius, n n. Next, allow the 
requisite margins for laps, 
•earn, or lock, as by the dotted lines, and cut out by those lines. 
Lastly, cut open the remaining portion of the bisecting radius, to 
the centre, o : then will the sector, B D F o B, be the cover ; and 
the arc, A C Ey will be ecpial to the circumference of the rim, or 
hoop. 

Example. — A bevelled cover is to be constructed having a rise 
of 1\ inches, and to fit a cylinder of 12 inches in diameter : then 

.s = v/[('|)- + 1.25-] = C.1289 inches ; 
iZ=G.1280-|-:i=zG.3789 inches; and 

B n F, the length of the ilexible measure, 
1=3.1416(0.3789 X 2 -12) = 2.381 inches. 

Note. — The value of s in all cases coming under this problem, since it is 
the length of the hypotenuse of a right-angled triangle wliose base and per- 
pendicular arc given, may be found by mechanical means, thus : construct a 




PATTETvN CUTTING. 



365 



right-angle by draft on a plane, making one of the legs oqnal to half the 
initial or given diameter (in the last. example inches), and the other equal 
to tlie given rise or i>erpendicular of the triangle (in the last example 
1^ inches), then will the rectilinear distance between the extremities of the 
legs, opposite the angle, be the hypotenuse, or value of s, required; thus, 
either side of a right-angled triaiigle may be found, by mechanics, tlie other 
two sides being known. But I must not be understood' by this as encouraging 
a desire to avoid the extraction of the square root of numl>ers by arithmetic; 
without the ability to do that, the student will find himself ofce'u perplexed, 
and occasionally defeated. 



Prob. 30. — To construct a Pattern for a Bevelled Elliptic Cover 
of a given Rise^ to ft an Elliptic Boiler of given Diameters. 

D = transverse diameter of boiler. 
d zzi conjugate diameter of boiler. 
h =^ rise or perpendicular height of cover. 

Rule.— Construct a right quarter-section of an ellipse, A C B,hy 
rule, Prob. 15 or 18, making A B equal to v'CCi^)' + ^H, and B C 
equal to \^[(hdy^-\-h^^. ^-,_,.^,_ 
M^ke Cn=i\(P-p% r - 
P being the circum- 
ference of an ellipse 
whose semi-axes arc 
\fl(iDy + lr} and , 

the circumference of 

one whose diameters 

are D and d; and 

from the point n draw 

a line n B : then will 

A n B he the unit 

measure of the cover, 

and will contain one-fourth part of it. Allow, as by the dotted 

lines, the necessary margins for the edge and half-lock. 

A B and B C may be found by mechanics (see Note appended 
to Prob. 29). 

Peob. 31. — To construct a Bevelled Cover of a given RisCy to Jit a 
False-oval Boiler of given length and width, 

D zz: length or transverse diameter of boiler. 
c?=: width or conjugate diameter of boiler. 
h = rise or perpendicular height of cover. 
S = v/[(iD)- + /r] =B E, diagram. 
N=z{SX d) — D — Bm, diagram. 
h' = (dXh)-^D. 
JP=: y'[(^cZ)- -j- /i'-] =: 7n n, diagram. 

31* 




366 



PATTEKX CUTTiyO- 



Rule. — Constrnct a irctanglo, A B C D, makinjr A D canal 
to J A a»(l ^ B and i> m each eciurrl to ^^. Make /^ E equal to 
y^(^Z>)2 ^ ^2^ and w 71 equal to v'CI^O' + ^ ''"' ^'^*''V with the 

Mjuarc in position 
7:J 7/ »u one of the 
bladrs cuttinj; the 
jK^ints /i and D^ and 
tlj»^ other cuttinj; 
the i>oint w, chaw 
the hnes /.; 7 A 77 u. 
Next, wiih the di- 
ridcrs find a radius, 
o By that will eut 
the |)oints 7i ami n, 
aixl with o the cen- 
tn.\ describe the 
arc -B n : then will 7? n 11 E he the unit measure of the cover, 
and contain one-fourth part of it, less the alluwauee, as by the dot- 
ted lines, for the edge and half the lock. 

NoTK. — Wlun d = \n. A' = 3*, «nd F, K, nnd /? o _ >. ^^ aiir! F may b% 
foiin<l bv Mcrhiinui, as by nil*' givtn In Note api»«Mn!«(l to It-ob. 'jy. lu prmc- 
ticc, if A be tivkeu equal to Sal^t the rL>^e will gcuirally be vuilkicut. 




Of Can-Tops. 

Can-Tops arc pimply truncated cones, and the c«»nis thiiiiM-lvrs 
arc pitched or l>evelled circles. They may Ihj defined in part by 
their pitchy whielj I shall here define to \)C the angfe f>f the si«lc of 
the cone to thci base ; or they may be defined l)y their bases and 
perpendicular heii:5ht. The Ixxly of a common tunnel is a two-thinls 
pitched can-top, or a ean-top havinjj a pitch of GO*' ; or, in other 
words, it is the frustum of a cone, or pitched circle, whose slant 
height was equal to the diameter of the base : it is therefore made 
up of a semi-circle whose radius is e(]ual to the greater base ; but 
can-tops are rarely pitched as steep as Go^. They may l>e con- 
structed in a single jnece, and slioulcl be when practicable; orthcj 
may be composed of two or more right-sections, as the Ixxly of a 
common tlaring vessel ; so they niay U; j>ieced transversely, when 
desirable, allcr the manner of ]>iccing a largi* tunnel. 



I 



Prob. 32. — To constntcf a i(iN-,nj) nj d fjicm Depth and jjiven 

Diameters. 

RuL£. — Proceed in all respects by rule, Prob. 1 or Prob. 2. 



PATTERN CUTTING. 



367 



Prob. 33. — To construct a Can-top of a given Pitch and given 

Diameters. 

D = diameter of greater base. 

d zn diameter of lesser base, 

1^= pitch, or angle of the side to the base, 

JF/= perpendicular height of generating cone, having D for its 

base. 
it = slant height of generating cone, having D for its bcise. 

r = slant height of cone, having d for its base. 



//: 



iZ) X sin V 



cos V 



= ^DX tan V=z y R' — HDf 



Dh 



'D — d 



R = 



\D _ H 



cos V sin F 



.^TP^^^Dy- 



cos V 



Rd 

'' D 



= y^H-hf + {hdf, 



lieight of the frustum. 

Rule. — From a common centi-e describe two concentric arcs 
of circles of sufficient length, 
taking R for the radius of one 
of them, and r for that of the 
other. Next, draw a radius 
from the centre to the outer arc, 
as r ic, diagram. Next, with 
a flexible measure, cut to the re- 
quired length, (the whole cir- 
cumference of one of the bases, 
when practicable), and bent to 
the proper curve, measure off 
that length on the curve, as from 
R to r, and from the new-found 
point, f, draw a radius to the 
centre, a : then will it v n r be 
the top, or a known aliquot part of the top, required. 




Special Cases. 

For a two-tMrds pitchy or pitch of 60° (sometimes erroneously- 
called a half-pitch, because the angle of the side and axis is half as 



368 PATTEKX CtTTTSG^ 

great as that of tlic side and base). — From a common crntro de- 
scribe two concentric semi-circles, taking the diameters of thC given 
bases, respectively, for the rad'ri : take the greater senii-cirele for 
the top required. 

For a half-pitchy or j /itch of 45^^. — Describe two concentrir 
circles, takiijg the diameters of the liases mnltipliei) by 1.4112, 
respectively, ior the raclii ; take three-fourths of the gre.iter circle 
for the body, less one-fourth of its ra<lins as a chord. 

For a flird'pifrh, or pitch of 30'^ (oHcn erroneously called a hall- 
pitch, because the angle of the base and side is half as great a.i 
that of the side and axis). — Fi*om a common centre descri!)e two 
circles, taking D X I.l-'^ID for the radius of one, and fl X l.l-'iin 
for that of the other. Next, draw a radiu;< from the centre to tlu- 
outer circumference. Next, take tiD-^v zn D-^\.Oi72 in the di- 
viders, and from the summit of the radius ju^t (Irawn, space it ofl* 
on the outer cinumference, <as a chord ; and from the last found 
point draw a radius to the centre : the greater sector, or sector of 
the re-entrant angle, will l)c the body or top required. 

For a quartcr-pilch^ (rr pilch of 22j*^. — From a common centre 
describe two circles, takmg 1.0824/) for the radius of one, and 
1.0824^/ for that of the other. Next, draw a common radius to tho 
circles. Next, take hi) in the tlividcrs, and from the |»oint where 
the radius cuts the eireumferenee of the greater cinde, step it olF 
on that circumference as a chord ; and from the last found point 
draw a radius. to the centre : take out the lesser sector, and the re- 
mainder of the circle will be the top required. 

NoTK 1. — Tho forc'ffoln;? ppi'dal casea are .ill cm-ered, of roiirse, bj the 

phcidinp ponrrnl rule; Imt It is iiion* ' ' = •= ih to measure an arc liy It8 

chonl, winii the l:itt«r Is known or rn: foiuul, tliun to measure it 

by till' Ili'xibU' nii'a>ure, nioro prnorn!!'. 

2. — Havin/x laid olT the di |;»ii<rn, as above directed, allow the 

requisite margins for burr U' • am, hh in other like cuios. 

.^. — A fiftk-pitrh, or yiltch oi i^ . i- -inietimeg, but incorrertly» called a qirar- 
ter-i)itoh, Ik'^'huj^i' the angle of the base and side i^j one-fourth 'a.s great a» thai 
of the axis and side. 

4. — The prevailing tendency of a can-top, or frustum of a cone, to '• lop at 
the lock,'' ns it is called, is charly due to the can-less manner of turning tho 
"wards and closing them. If the workman will turn the ward^ parallel to 
the linos of the dimensions pattern, and will use neither m«)re nor le.-s for the 
lock than the marginal he allows, the tendency cor!ip!:iined of will not obtain 
(see Note, I'rob. 1). 

Ok ijii> i »Mi. ^*ii-. A^» iU.r>. 

Lips for measures, when laid off on a plane, are simply lunes, or 
crescents. In practice, no arbitrary relations between the lip and 
the vessel it is intended for are sought to be maintained. Thus, 



PATTERN CUTTING. 869 

the length of the lip, its width, and its angle with the side of the 
vessel, are matters of taste or choice with the workman, limited, of 
course, by the purpose the lip is intended to subserve. 

The following rule, Prob. 34, which sets the lip at an angle of 
41° 50' to the side, very nearly, is the one most commonly in prac- 
tice ; and it appears to be as good a general rule as can be offered. 

Prob. 34. — To construct a Lip for a Measure^ the Diameter of 
the Top of the Measure being given. 

Rule. — Take three-fourths of the diameter of the top of the 
measure in the dividers, and with that as radius describe a circle. 
Kext, with the same radius, from a new centre, taken about mid- 
way between the centre of the circle and the circumference (more 
or less, according to the desired width of the lip, wired-edge includ- 
ed), describe an arc cutting both sides of .the circumference : then 
will the crescent thus formed be the lip intended. 

Note. — To diminish the pitch, which serves to make the lip longer, take 
the radius at a greater ratio to the diameter of the lesser base, or top, than 
3 to 4. But, even when the measure has little or no flare, the radius should 
not much, if any, exceed seven-eighths of the diameter of the top. 



Of Sheet Pans. 

Sheet pans are vessels intended to hold fluids at a temperature 
above the melting point of solder. They are constructed of a sin- 
gle rectangular sheet, by turning up the sides, and folding the 
surplus surfaces at the corners upon the sides. 

Dripping-pans and baking-pans are commonly constructed with 
sides slightly flaring, while evaporating-pans, most generally, have 
perpendicular sides. 

In geometry, those having oblique sides are called prismoids, or 
prismoidal vessels, and those having perpendicular sides are called 
parallelopiped vessels, or prisms. 

These vessels are commonly constructed with wired tops, or rims, 
partly for the purpose of stiffening them, and partly to hold the 
sides in place. 

Sheet pans maybe constructed to given dimensions, — length, 
breadth, and depth, — and thus to given capacities ; so they may be 
constructed to given ratios of parts ; but, generallj^ economy of 
stock and utility of purpose, without further specifications, are 
allowed to govern. 

With these remarks (and the additional one, perhaps, that the 
sides of a sheet pan are to be of equal width), we might dismiss 



870 PATTERN CUTTING. 

the subject, were it not that rules for cutting the comers so as to 
bring the edges up to, or under, the wire, seem to be called for. 

PROB. 35.— To cut the Corners for a Perpendicular-isiJed Sheet Pan, 

Rule. — Let A B and D E^ diagram, represent the width re- 
quired to cover the 
wire ; B C and D C, 
the width of the 
sides. Itemove tlie 
section A B m 1) K : 
then will the stpiare 
B m I) C\ fohled di- 
agonally, Jind re tie X- 
eu u|)on the side or 
end, form the c^lge 
pro|>osed.* 




Prob. 36. — 7o cut the Corners for an Ohl'mue-s'ule.d Sheet Pan. 

Rule. — Let A B and D E, diagram, n^present the width or 
margins riMjuirod to cover the wire; /> (^\ind /) C\ the width of 
the sides ; 1* h and 7) J, the Hare of the sides. Draw /; u perpen- 
dicular to a C, and d n perpendicular to c (\ Next, remove the 
section n h n d e: then will the trapezium b C d n, folded diago- 
nally from n to C, and reflexcd upon one of the sides, form the 
edge contemplated. 

Prob. 37. — To construct a Hearty or Hcarl-shaped Cake-cutter, 

Rule. — Construct the Gothic arch, No. 1 ; this work, and upon 
its ( hord, describe two equal semi-circles, in diameter equal to half 
the length of the chord. 

Prob. 38. — To construct a Mouthpiece for a Sjieaking-tuhe. 

The curve for this structure consists of four cymas of unequal 
branches, in pairs. The structure itself is little else than two 
short, decidedly llaring teakettle spouts, in a single piece, having a 
side common to both. 



PATTERN CUTTING. 



a7i 



pROB. 39. — To construct a Pattern for the Body of a Circular- 
bottomed Flaring Coal-liod^ all the curves to he arcs of circles, 

D= nominal diameter of greater base. 
dzzi nominal diameter of lesser base. 
h =: perpendicular depth of vessel. 



D-d' 



D 



Rule. Place the square on the plate and scribe to its edges, as 
A t 0^ diagram ; produce A t in the direction B, and o t in the 
direction D, making all 
the lines from t of suffi- 
cient length. Next, : 
make t B equal to one- 
fourth part of the cir- 
cumference of the 
greater base =0. 7854Z), 
and make t A equal to 
t B. Next, make t oz=:H, ^ 

— Dh-^(D-d), and ' - . 

draw the radii o A, 
Bz=R, Next, drop 
the square perpendicu- 
larly down on the line 
t 0, equal to the given 
perpendicular depth, /^ 
as from t to z, and 
through the point i draw 
the line in i n, parallel 
to the line A t B. Next, 
take the nominal diam- 
eter of the greater giv- 
en base (i)) in the 
dividers, and with one 
foot in the point B^ and 
the other in the line 
t 0, as at /, and / the centre, describe the arc B D A. Next, with 
A the radius and o the centre, describe the arc A C B. Next, 
with ?n, zzz r, or o n the radius and o the centre, describe the arc 
m k n ; then will A D B nk m be the front half of the body, or a 
pattern covering that portion ; and A C B n k ?«, in a separate 
and distinct piece, will be the Iback half of the body, or a pattern 
covering that portion. Both portions are introduced here in the 





372 PATTERN CUTTING. 

same diagram, that their relations may be perceived and readily 

comprehended. 

Note. — Tliis style of hod is introduccnl, partly to me«t the popalar demand 
with regard to the spout, partly with reference to economy in stock, and partly 
because of the readine.«s with which it may be plotteil upon a ulauc surface, 
compared with the labor of plotting for an oval or elliptitMil t>ottom. The 
workman will find no ditriculty in constructing it, except, perli;i rtiini^ 

the rim for the wire in the immediate vicinity of the si(ie-Io< r will 

probably be obli^e«l to do on the stake. When lockni at the m \ired, 

It is to be compressed along tlie up|>er front half, and al>o in Iruui, so as to 
form nearly a perpendicular-sideu spout, of about four ioche8 in width at 
th<- lip. 

The real bases of the vessel will l>e somewhat greater than the nominal, 
because the chords of the arcs are made e<pial to tlie given half-circumferences, 
instead of the arcs tliem>elvcs ; the diameter of the bottom, therefore, will ha 
eciual to twice the arc m k- ;i, divided by :i.l4ir», insteatl of iK'ing equal to ti. 

In practice, If the nominal diameter of the greater bas«' Ik* taken e^jiml to 
once and one-half that of the lesser, or the nominal dian ' • • •,.!. \jq 

taken equal to two-tbirds that of the forunr, and the \h ;I» be 

taken eoual to tlie nominal diameter of t'lo li--fr r>a-< - will 

b*; foun<l satisfactory : moreover, if tlw n > aur base 

be taken at 12 inchrs and llie fongoin i very fair 

medium-sized hod will be obtained, pan; : . laargins bo 

allowed. 

The hoop for the bottom, which ehouM ' :i) tit one and a half inch In 
width after it is wired, may have the sana- ilai' i Mm- IkxIv, and the radius of 
itfl lesser arc will be the same as that for the U —' v ba-r ui the body. 



SOLDERS. — ALLOYS AND COMPOSITIONS. 373 



SOLDERS, ALLOYS, AND COMPOSITIONS. 

Hard solder. — Copper 2 parts, zinc 1 part } — used with powdered borax. 

Pewter er''s solder. — Tin 2 parts, antimony 1 part. 

Tinman's solder. — 1 part each, lead and tin. 

Plumber^s solder. — Tin 2 part, lead 5 parts *, or, pewter 4 parts, tin 1, and bismuth 1. 
Ilesin is used with the last three. 

Solder for iron. — Tough brass, used with borax. 

Silver solder. — 1 part brass, and from 2 to 5 parts fine silver. 

Spelter " for brass, copper, and German silver. — 2 parts brass, 1 part zinc. 

Solder for copper. — Brass 6 parts, tin 1, zinc 1. 

Dentist's solder. — 4 parts 22 carat gold, 1 part silver, 1 part copper. 

DentisVs gold. — 10 parts 22 carat gold, 1 part silver, 1 part copper. 

DentisVs compound for clasps. — 5 parts 22 carat gold, 1 part platinum. 

Yellow brass. — Copper 3 part?, zinc 1 part. 

Spelter. — Copper 2 parts, zinc 1 part. 

For lathe bushes. — Copper 16 parts, tin 4 parts, zinc 1 part. 
" " " harder. — Copper 16 parts, tin 4 parts, zinc 2 parts. 

Improved Babbit metal. — This composition, for the lining of boxes, shaft bearings, &c., 
— which, from the satisfaction it has thus far given, bids fair to come into general use, — • 
is composed of tin 12 parts, antimony 3 parts, and copper 2 parts. The original recipe 
for this alloy was, tin 6 parts, antimony regulus 2 parts, and copper 1 part, as its prime 
equivalents, to which, when about to be remelted for use, 2 parts of copper to 1 of the com- 
position were added. 

For pulley blocks. — Copper 7 parts, tin 1 part. 
" wheels^ boxes, and cocks. — Copper 8 parts, tin 1 part. 

Bronze — government gun-metal. — 9 parts copper, 1 part tin. The specific gravity 
of this composition is greater than the mean of its constituents. 

For valves. — 10 parts copper, 1 part tin. 

Bell metal. — 39 parts copper, 11 parts tin. 

Gong metal — 40 parts copper, 5 tin, 2.8 zinc, 2.15 lead. 

Bath metal. — 32 parts brass, 9 parts zinc. 

Blanched copper. — 16 parts copper, 1 part arsenic. 

Britannia metal. — 1 part each, — brass, tin, bismuth, antimony. 

Petong, or Chinese white copper. — 20.2 parts copper, 15.8 nickel, 12.7 zinc, 1.3 iron. 

German silver. — 2 parts copper, 1 nickel, 1 zinc •, when intended to be rolled into plates, 
it is composed of 60 parts copper, 25 parts nickel, 20 of zinc, and 3 of lead. 

Manheim gold. — 3 parts copper, 1 of zinc, and a small quantity of tin. 

Mock gold. — 16 parts copper, 7 parts platinum, and 1 part zinc. 

Mock platinum. — 8 parts brass, 5 parts zinc. 

Speculum metal. — 7 parts copper, 3 zinc, 4 tin ; or, 6 parts copper, 2 of tin, and 1 of 
arsenic. 

Tombac, or gilding metal. — 9 parts copper, and 1 part zinc. 
Mock iron — expanding alloy. — Lead 9 parts, antimony 2 parts, bismuth 1 part. This 
composition expands in cooling, and is used in filling small defects in iron castings. 

Ring, or jeweller''s gold. — 150 parts pure gold, 39 parts copper, 22 parts pure silver. 

Queen's metal. — Tin 9 parts, antimony 1, lead 1, bismuth 1. 

Pewter, common. — Tin 4 parts, lead 1. 
" best. — Tin 100 parts, antimony 17. 

Steel alloyed with ^^ part of platinum, or to the same extent with silver, is rendered 
harder, more malleable, and better adapted for every kind of cutting instrument. 

32 



374 SOLDERS. — ALLOYS AND COMPOSITIONS. 

Solder for sold. — 3 part<? gold, 1 part pilrer. 1 part copper. 

Solder for Britannia. — Tin, 7 l)a^tJ^ : lead, 4 parta. 

Yellow solder. — Copper, 1 part ; zinr, 1 part. 

Black solder. — Copper and zinc, carli 8 parts ; tin, 1 part. 

Peivterers soft solder. — Bii>niUth, 1 part; tin, 2 part*? : Irad, 1 part. 

Common Britannia metal. — Tin, lOj partfl ; copper, 2 partfl : aDtimony, 1 part. 

Common bronze iiutal. — Copi>er, 4 parts ; zinc, 2 ports : tin, 1 part. 

Whitf 7?ietal. — 5 parU copixT, 3 rinc, 1 lead, 1 tin. 

Silver-colorrd metal. — Tin, 00 parta ; copper, 8 part* ; antimony, 8 porta; bta- 
muth, 1 part. 

hnitdtion .'^ilrtr. — IC parta copper, 1 part zinc. 

Pinchbrrk. — Copper, 4 parta ; zincj 1 part. 

Mtial for tdk-itif; Iwjirr.ssions. — Difimuth, partfl: lead. 2 parts ; tin, 1 part. 

Hii-rt metal. — ('oj>i>er, 10 parts ; tin, 5 |Mirtii ; zinc, 2 part«. 

Fusible alloy (tnclU at 200**). — ]{i.«Qiuth, 2 |>artM ; load. 1 jvirt : tin, 1 part. 

Muriate of zinc. — Muriatic acid, liolding in i>o]ution all the zinc it will dlwolTe. 

Acid for sold I ring tin. — Miiriat*? of zinc, 1 part by volume; poft water, 2 i»artii 
by volume ; add a trifle of J<al. .\imnoni.ic. 

Arid for soldering; ztnc. — MuriaU; of zinc, 10 ounce* ; Sal. Ammoniac, 1 ounce ; 
water, 1 pint. 

Actd for soldering brcLss or copper. — Muriata of zinc, 6 parta ; Sal. Ammoniac, 1 
purt. 

Ari^i for soldering gold or silver. — Muriatic Acid, 2 parts ; sperm tallow, 1 part ; 
Sal. Ammoniac, 1 i>art ; all by weight. 

Acid for .^oh/trini; iron. — .Muriatic Acid, IC parts; SQcrm tallow, C part« ; Sal. 
Ammoniac, 4 purls; all by weight. 

Tinning arid, for brass or copper. — Muriate of ilnc, 4 parts ; soft w»tor, 4 parts ; 
Sol. Ammoniac, 1 part ; all by weight. 



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